For my understanding artistic works get copyright from the moment of their creation. This would allow one to pick battles based on how lucrative they may potentially be.

In the US, you need your copyright to be registered in order to file an infringement suit or be granted statutory damages. This must be done prior to the infringement, so they wouldn’t be able to pick and choose which to register after the fact. The fact that (unregistered) copyright arises from the moment of creation is true, but not particularly useful here.

You dont really need art museum of babel for this but you just tons of different works that may contain unique characters, structures or objects similar to what someone might be able to imagine or has already imagined.

Copyright is not the same as patents or trademarks; someone coincidentally creating something very similar or even an exact replica of your work is not infringement.

If whether you copied from their work or independently made similar choices is under question - then close similarity of the works could skew the balance of probabilities. However, the courts will be able to see that coincidental similarity is far more likely if a colossal number of images have been registered.

You may draw fan art with disney characters but its actually illegal to sell said art work without Disneys aproval until copyright expires.

It’s still copyright infringement even if you publish it non-commercially, but a Fair Use defense would likely hold up.

To file an infringement suit they’d need to have paid registration for each work which, even for the exorbitantly rich, wouldn’t be remotely feasible for all logical arrangements of words/images. There’s probably not even enough space in the Universe or time until its heat death to generate and store all such images.

Even if they did, copyright doesn’t protect against against independently created works that happen to be similar or even identical - so they wouldn’t be exhausting some limited set of possible works by doing so.