Painting, too. Not sure how common or uncommon but my landlord explicitly allows painting, as long as it's not a dark color
Yes; but matching the paint color? and matching it to the wear of sun/fade of the surrounding area? That's not nearly as straightforward.
Is there a law saying your landlord has to accept the repair?
In most jurisdictions, what you can take out of a deposit is legislated.
And it's really not that hard to make an unnoticeable drywall repair.
Not sure, but they likely won't know. Too many people ask permission for things. You'd have better luck doing what big corporations have done for decades. You just do the thing, cover it up, and hopefully never have to ask for forgiveness. And if you DO have to ask for forgiveness, well that's still easier than asking for permission!