First let's look at just Mirage Arcane on its own, to get a baseline, and then we'll look at what Illusory Reality adds.
An illusory pit is not actually underground!
Mirage Arcane only changes the feel and appearance of the terrain, not its actual shape.
You make terrain in an area up to 1 mile square look, sound, smell, and even feel like some other sort of terrain. The terrain’s general shape remains the same, however. Open fields or a road could be made to resemble a swamp, hill, crevasse, or some other difficult or impassable terrain. A pond can be made to seem like a grassy meadow, a precipice like a gentle slope, or a rock-strewn gully like a wide and smooth road. Similarly, you can alter the appearance of structures, or add them where none are present.
Every part of the spell changes the experience of the terrain, not the actual terrain itself. A stick can be picked up — but only because it feels real and interacts with you magically-intelligently. A precipice can appear to be a gentle slope — but you will still fall vertically if you try to walk down it. Difficult terrain can feel difficult to move through — but to a viewer with truesight you're just moving yourself unnecessarily slowly to deal with non-existent obstacles. In all cases, the terrain alters your belief about what you're interacting with (and thus alters your behaviour accordingly), not what you are actually interacting with in reality.
So: an illusory pit. The pit would be visible, and if you stepped above it, you would feel, see, and appear to others to fall into it. You would still be on level ground aboveground though, in reality. When the illusion fades, you would not be somehow underground, as the spell does not have the power to change the terrain in a real way and you never were below the ground in the first place.
Now add Illusory Reality
The Wizard class feature that lets you pick one object in the illusion to make real is a bit tricky.
A stick could be made real by weaving shadow magic into it, and then it would have real substance, not just trick you into believing that it has real substance. An Illusory Reality stick could help you dig a hole in the ground, and the hole you make will still be there after Mirage Arcane ends. Without Illusory Reality's shadow weave, you could still dig that hole, but it would be only an illusion and mistaken belief and disappear when Mirage Arcane ends.
So, Illusory Reality can make what you believe, see, feel, hear, etc. happening when you interact with an illusion into an actual fact. Walking over a chasm created by Mirage Arcane would result in believing you're crossing the chasm, at least until you hit bottom and the awful reality literally hits you. With Illusory Reality applied to that bridge, thinking you're actually walking across it would match the reality: you would cross the bridge and not fall, because the shadow weave has made it real, unlike the rest of the illusion.
How does this apply to a pit? The trouble with a pit is that it is not an object. As the joke goes:
Q: What becomes larger the more you take away from it?
A: A hole!
A hole is not actually an object, even though it's convenient to give it a name and think of it as an object sometimes. Investing empty space with shadow weave isn't going to work because there's nothing there to weave it into — it's not an object.
Not convinced? Look at it from the other direction and consider where the dirt would go. Does weaving shadow stuff into the space where the dirt is make the dirt suddenly unreal? No, Illusory Reality doesn't have the power to unmake things, only to temporarily make things.
So, you can't make an illusory pit real with Illusory Reality.
What about buildings?
Sure, you can make a fake building from Mirage Arcane into a temporary shadow-stuff actual object. The stone and timbre of its walls will actually support a character's weight, instead of just making them believe they are, just like with the chasm bridge. (And just as with the bridge, they can fall when the spell ends and the shadow-magic stops supporting them.)
However, underground constructions would still not actually put a character underground. The ground is still not "unmade" by either the illusion or by the Illusory Reality. A character could still think they were moving around in the below-ground spaces of such a building, but in reality they would just be walking around above the ground, on the still-existing actual ground, and the below-ground shadow-stuff would become irrelevant, buried inside the real ground. When the Mirage Arcane wears off, regardless of Illusory Reality the character would still be aboveground.
Yes, within some restrictions.
There are a wide range of illusion spells, each of which have their own specific conditions and limitations. This means that we have to look at each spell in turn. Generally, though, you should be able to create at least some of the effects you're looking for.
Illusions on creatures
Creating moving illusions on creatures is much easier, and there are several spells that allow for this. Specifically, spells like Disguise Self and Seeming allow you to change the appearance of your body, clothing, and any items you carry, including weapons: (text from Disguise Self)
You make yourself, including your clothing, armor, Weapons, and other belongings on your person, look different until the spell ends or until you use your action to dismiss it.
In fact, the text of disguise self gives an example of an illusory hat:
For example, if you use this spell to add a hat to your outfit, Objects pass through the hat, and anyone who touches it would feel nothing or would feel your head and hair.
Thus, altering your weapon to look scarier or adding sweaters to dogs is perfectly plausible with disguising illusion spells.
Illusions on objects
It's a little more difficult to make an illusion that moves on objects. You could accomplish everything you ask about with Major Image, but you have to spend an action on moving the image around:
As long as you are within range of the Illusion, you can use your action to cause the image to move to any other spot within range. As the image changes location, you can alter its appearance so that its movements appear natural for the image.
However, because you can have the illusion alter itself to appear natural during the movement, you can alter the sweater on the dog to move like a real sweater or have the vase wobble on the table. This is a rather costly spell to use, given that you have to be in range and constantly spend actions in order to maintain it. Note that Minor Illusion doesn't say that you can move it around with an action, so you can't use it to create these effects.
You could use a Programmed Illusion to replicate some of these effects, but given that the illusion is fully scripted, it doesn't seem like the kind of dynamic response you're talking about.
Illusions on Terrain
For spells like Mirage Arcane, the terrain itself is changed in appearance. This means that if the terrain moves (as the world turns, for instance), the illusion will also move with it. It's worth noting that the world isn't always round and rotating, but that's usually up to the DM. Generally, the rules don't specify a particular frame of reference for illusions.
Best Answer
As mentioned in the comments, this is largely a DM call, but here are some thoughts to help adjudicate this based on RAW and official rulings (and some of my own musings).
First, the illusion can be broken into two categories - changes to the landscape, and decorations placed upon it.
Landscape
As the spell states (RAW), you cannot change the general geometry of the landscape - flat land (or water) remains flat in the illusion, and vice versa. So the "height" of this component of the illusion would be no more than a few inches, a couple feet at most. It is important to note, however, that the official rulings seem to contradict this - it allows the creations of a crevasse, though it doesn't specify a rough size for this feature - it might be a mere crack in the ground a few centimeters wide. They also allow for the creation of cliffs, though again, the cliff may follow an existing feature - turning a 3 story building into a sheer rocky cliff, for example. The wording in these developer comments implies that the change (the cliff and crevasse in these examples) is a change in the geometry of the landscape, and not just a "re-skinning" of an existing feature, but does not explicitly state that, so DM beware.
The best way to adjudicate this might just be to apply the geometry restriction in only the vaguest sense, which is to say that local exceptions can be allowed so long as the general character is unchanged. A lake can be turned into a plain, but not a large hill. In the middle of that plain, you can have a burbling brook that has carved a 4 foot deep crevasse. This doesn't conform to the existing geometry precisely, but taken as an average over the whole lake, it doesn't move the needle too far.
How you handle interaction with such things as an illusory hole in solid, real ground is dealt with in other questions. I recommend looking to Phantasmal Force.
It's also worth discussing here that the spell contradicts itself in this regard. It states that "the terrain's general shape remains the same", while immediately including and example of changing the appearance of "a precipice" to one "like a gentle slope". So, it seems that at the very least, some liberty can be taken here with the shape of the landscape, if not simply ignoring the restriction entirely. I guess it impels the question of what exactly they mean by "general shape". We have largely assumed it to mean the altitude since that seems to be the general direction of the examples, but it might be worth considering that it may mean something else, though I struggle to imagine what. My best guess is that they are simply stating the (what I feel is) obvious - that while the illusion could be anything, the actual physical terrain is not changed by the spell. This interpretation eliminates a considerable amount of confusion and makes the spell simpler to use and adjudicate.
Decorations
These are even less well defined, though examples given include buildings and trees. No guidance is given on how big these can be - can you place a giant redwood or a towering castle? By RAW, it appears so.
There are a couple ways you can adjudicate this (well, there are infinite ways you can, but I'll cover two).
One, you can take the "1 mile square" area affected, and simply extend that vertically, allowing any decoration (outside of geological features and creatures as they are specifically limited elsewhere in the spell description) up to that size. A Costco that is a mile high and a mile on each side? Done. Jack's Giant Beanstalk, extending to the clouds (those that are lower than 1 mile, anyway)? Done. A mountain? Nope, sorry (could you even fit a decent one in a mile square?).
Two, you can place some arbitrary measure of "reasonableness" on the decorations. Nothing fantastical. Nothing "too large". So, no floating castle on a cloud. In fact, no castles at all as they are "too large". It would be up to the GM what "too large" is. Maybe nothing larger than a merchant shop, maybe 20 feet to a side and a single story tall. No trees taller than 20 feet. That sort of thing.
Don't hurt your brain
House rules ahoy!
If you try to create hard rules, it might be less fun. It might just be better to allow most anything with the caveat that it should not emulate the effects of combat spells or non-illusion spells, short of the general solidity of the illusion and the changes to terrain difficulty. It also seems that emulating the effects of Phantasmal Force (an illusion spell, but a combat spell as well) is allowed as the objects created by MA can cause damage. So, maybe just allow any illusory effect (combat or not) that doesn't run afoul of the restrictions on creatures (no adding, disguising, or obscuring).
In short, MA can be thought of as a Phantasmal Force spell that affects everything in a mile square (as well as those that can see that area), with the added benefit of creating semi-solid interactable objects and changing the difficulty of terrain, and with the limitation that it can't create illusory creatures, nor obscure or disguise real creatures. Being partially real (unlike PF, which is completely within the victim's mind), it is also detectable by truesight. It also has the benefit of not allowing a save, so it will always be "believed"
Under this interpretation, if the caster wants to create some fantastical land of rainbow rivers and levitating buildings, he can put a "Welcome to Fun Land" sign on it and sell tickets. But very few people will believe the illusion (which probably isn't the intention for that particular casting anyway).
If he wants to use it to create a vast underground cave complex, let it be. Is it some extra-dimensional space or a massive Holodeck trick where you just think you are descending into the earth but never leave the surface? Don't worry about it. When the spell ends, everyone that was in the tunnels ends up on the surface anyway, and it doesn't matter if they were there all along or not. Unless the underground caverns were supposed to protect the town's inhabitants from the destined meteor impact that would wipe out the city, in which case, put on your DM pants and figure out if you want it to work or not (I'd probably go with "no").
Also under this interpretation, if they want to turn a lake into a large hill, let them do it. I don't think it unbalances the spell, though it is certainly not RAW. My best guess as to why the RAW restriction against changing the landscape's geometry is there was to prevent the spell from being used to deny access to an area by burying it under 300 feet of imaginary soil or putting a 500 foot deep valley beneath it and forcing some ability to fly be used to access the object now in midair. However, the spell as written can effectively hide or disguise objects anyway (it is only restricted from hiding or disguising creatures), and it is easy enough to rule that willing yourself through the illusion (once you realize it as such - basically only with true sight as there is no Save listed for the spell) is trivial, preventing the spell from hindering those that see through the illusion, though they can continue to interact with those aspects of the illusion they don't will themselves to ignore, because the spell specifically states that it allows interaction even for those that know its nature (emphasizing the "can" in "can still interact" which means that the creature can also choose not to interact).
The Landbridge
Considering the other MA question ("Can I use the Mirage Arcane spell to walk on top of a stretch of ocean?"), it might occur to some to imagine that if there were no limit to the height of imaginary items placed on the imaginary landscape (a potential third way to answer this question) that you could place a 500 mile high tree within your MA area, then cut it down and then use it as a bridge between two distant continents. However, the spell specifically prohibits this as "any piece of the illusory terrain (such as a rock or stick) that is removed from the spell's area disappears immediately." This includes things that are only partly removed (though "any piece" could be interpreted differently, it makes sense that the mile square is an area of effect and that the illusion, or any part of it, ceases outside of that area), so anchoring one end of the fallen tree within the MA spell doesn't allow the rest of it that is outside the spell to remain solid. Whether only the portion of the tree that leaves the area of effect disappears or the whole thing does is again, left to the DM.
The Answer
Could the spell displace water? The water is not part of the landscape at the bottom of the ocean (or the bottom of a lake, for that matter), it is more like the atmosphere. So I don't think it would.
Could you create a building with an airlock in it? Yes, but the inside of the building would still be filled with water.
What about a bubble of air as a floating decoration. Hmm, maybe floating decoration should be disallowed, you think. But the bubble could just as easily be attached to the "ground", so that still doesn't solve the problem. Of course, a bubble is kinda like a hole in the water, so we get back to the whole "is a hole an object" debate.
Anyway, we can look to Phantasmal Force to adjudicate this. You can totally shove your head into that bubble and breathe deep, but you will still drown. Your mind will fill in the blanks to cover the inconsistency - that damn bubble popped before you could get enough air out of it, or it turns out the pressure at this depth is preventing you from inhaling properly, or the bubble turned out to be filed with some non-oxygenated gas. Whatever.
Bottom line, you still drown, and you walk (move) at half speed as underwater is not considered difficult terrain if you have no swim speed - it has its own rule that states that each foot of movement costs 2 feet of move, so the fact that the spell specifically allows you to ignore difficult terrain if the illusory terrain is not considered difficult is irrelevant. Likewise, the BBEG would move freely, and he would somehow (from his point of view) be able to breathe normally (assuming he breathes underwater) if the caster created an illusory space that appeared air filled.