Can Nolzur's Marvelous Pigments be used with the Mage hand cantrip?

Question

I am trying to figure out a way for a 1st level wizard to escape from a cage.

This is the scenario:
he is trapped without any equipment (just a simple robe, and no spell left, except his cantrips: Toll the Dead, Dancing Lights, Mage hand) into a cage with iron bars and an iron door closed with an iron padlock (visible, but not reachable from within the cage), but the rest of room is empty, except for a chair and a table that is 15 feet away. On the table he can see a small paintbrush and a pot (that he knows is the valuable Nolzur's Marvelous Pigments) chained to the table.
No guard will be around to watch/hear him for the next 45 minutes.

1. Can the wizard cast Mage hand and use the paint brush to paint a vial of acid (that, if I am not wrong, should cost less than 25 gp) to turn the iron padlock into rust? I am asking since I know that the Mage hand cannot activate magical items, but should the act of "painting" be considered a magic item activation?

2. Does Mage hand possess enough agility to paint the vial? And then to pour/crash the vial on the padlock?

3. If the creation of the vial takes more than a minute, can the wizard cast Mage hand multiple times to go on painting it to complete the object creation? How long can such creation take?

4. In case one vial is not enough, if the value of the acid is ~20/25 gp, how many vials can be created with a full pot of Nolzur's Marvelous Pigments?

5. Am I forgetting any easier way for the wizard to break free? For example: could be more acceptable for the DM that the wizard use the Mage hand with the paintbrush to simply splash the magic pigments onto the padlock with the desired effect to create the acid directly on it?

Answer 1

Probably yes, but ask your DM

The crux of this is if using the pigments counts as activating a magic item. There is a Q&A that claims using the paint does not count as activating a magic item. I’m not sure about that. If your DM decides it does not, as that counts as activating a magic item, then the story ends here. You cannot paint with them.

So let's assume instead that it works.

Mage Hand then says

You can use your action to control the hand. You can use the hand to manipulate an object, open an unlocked door or container, stow or retrieve an item from an open container, or pour the contents out of a vial.

There is no statement about fine control, but you also do not even need that. Marvelous Pigments says:

The paint flows from the brush to form the desired object as you concentrate on its image.

No fine control needed, you just need to concentrate. So, if you can use the pigments, you can paint vials, one by one, many of them, and can pour them on the lock to rust it off.

Answer 2

Whether marvelous pigments can be used by mage hand is up to the DM

It's dubious. The hand can't "activate magic items", and using the pigments seems a lot like activating a magic item. A cooperative DM might see the pigments as an exception, since their description never mentions a specific action other than painting.

But you're the DM. You can allow it to work. If you do allow it to work, prepare yourself for potential future shenanigans as your wizard (no doubt locked up for shenanigans in the first place) looks to find ways to use the pigments at a distance for fun and profit.

Maybe the hand could get the pigments unchained

Maybe these idiots¹ used one of those chains like they used to use in banks to chain the pen to the counter, and the hand can just break or undo it. Or maybe the hand can find a paperclip and use that to undo the screw holding on the chain.

But what would be the fun in that?

If it did work

The acid is unnecessary. The mage could simply paint a break in the chain, and then the mage hand could fetch the pigments. Then the pigments could be used to paint breaks in the bars, or paint the lock away:

. . . painting a door on a wall creates an actual door that can be opened to whatever is beyond. Painting a pit on a floor creates a real pit . . .

If you can paint a door or a pit, you can paint away the lock.

For extra fun, once free the wizard can paint the bars back, then paint a spilled and dried up marvelous pigments where the one they liberated was. Then, it's not even obvious what happened.

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^{1 - I mean, "prisoner in cage" and "magic item that can create just about anything" . . . it seems sort of obvious that those two things shouldn't be stored together. I guess the room where they keep their lit torches and barrels of oil was full.}