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.