[RPG] Are Illusionists subject to their own illusions

No, Magic Missile does not hit the target. It sails right through it. Magic Missile targets a creature, and in your mind the illusion is a creature.

Each dart hits a creature of your choice that you can see within range.

The end result is that the Magic Missile is cast, the slot is expended, but it does not hit anything unless you specified more than one creature and any of those other creatures are valid targets (non-illusionary creatures). This may give the caster reason to believe his target is not quite what he expected, and may give cause to inspect the Silent Image more closely.

As for what actually happens in the narrative when this occurs?

Let's say the caster casts magic missile and decides to hit one (unknown to him) illusionary target and two other real targets, one missile each. The missiles streak towards their targets, and hit the two real ones. The third target is an illusion and can't be hit, so the missile passes right through him. This may tip the caster off that this may be an illusion. This is the most likely scenario.

The other option is for the DM to say, "that one isn't a valid target. Try again." This, of course, tips the player off, and lets the player choose a new target for his missile. This way involves metagame information, pulls the players out of the scene, and is probably a lot less fun.

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.