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.
Clever, but unlikely
In order for Catapult to work, it requires:
one object weighing 1 to 5 pounds within range that isn’t being worn or carried.
This is the requirement that Delayed Blast Fireball must meet. The question is: Does it?
Delayed Blast Fireball isn't explicit in the definition, but it does state:
A beam of yellow light flashes from your pointing finger, then condenses to linger at a chosen point within range as a glowing bead for the duration.
How much does a bead of light weigh?
In order for that bead to qualify for Catapult, it must be at least one pound and less than five pounds. As Dale M states, that's likely a DM call. But how should the DM approach?
Well, it's a beam of light that condenses into a bead. How much does a bead of light weigh? I'd think that a bead of magic light does not weigh anything, but another DM may believe otherwise. Then it's a question of does a bead of light weighs more than 1lb. Even if it did have a mass, it's unlikely to weigh more than 1lb.
Without meeting the weight requirement, you can't use Catapult to launch the Bead for extra damage (pending the save failure).
Throwing the Bead
There is another option for interacting with the Delayed Blast Fireball:
If the glowing bead is touched before the interval has expired, the creature touching it must make a Dexterity saving throw. On a failed save, the spell ends immediately, causing the bead to erupt in flame. On a successful save, the creature can throw the bead up to 40 feet.
This does imply that it can be handled, but it doesn't answer weight or anything else. Ultimately, this is a magic bead made of condensed light and it does not necessarily mean it has mass. It just means that you can throw the magic bead. If a DM does rule that it has mass, then it still needs to weigh more than 1 lb (see discussion above on weight of condensed light.)
Knock Yourself Out (and Don't Knock a Creature Out)
As the spell says emphasis mine:
The object is the target and the spell launches it up to 90 feet. It goes in "a direction you choose". You choose whether to launch it in the direction of one or, ideally, more creatures (in a line up it would go through each one until one failed the saving throw). It only takes damage and possibly breaks by the terms of the spell (rather than a DM ruling on falling to the ground or what have you), if it hits something (creature, door, whatever).
Note that the "up to" implies you also presumably choose the distance it goes (or maybe not, for a thorough discussion see this question). In any case, it helpful for snatching an object in range and stopping it wherever you want. Catapult is a utility as well as an attack spell. This is, in fact, one of the principle (mechanical) reasons one might choose this "might hit and requires an object" spell over something like the "guaranteed to hit for an average of three less damage" Magic Missile (I believe the other popular reason is that you can fire a flask of acid or such, potentially for more damage on a hit).
In any case, if you ever have that occasion when the McGuffin is on a pedestal next to the big bad rather than held, you'll have a lot of fun with this spell. It's also, naturally, what a stylish wizard would use to throw their grappling hook.