If you are open to using a clay instead I think that might create a better lasting piece for you. For a thicker salt dough, you just need to make sure you bake long enough but use a very low heat. I would air dry first both sides for one day each to cut down on the baking time.

Yes you can completely air dry it, but make sure it’s not too thick.

Can this dough air dry?

That should be fine, just make sure you refrigerate and allow it to come to room temperature for quite a while before you try to roll it out.

Yes that should be fine. You may need to get an acrylic paint marker if the permanent marker does not write nicely.

A little bit should be fine!