How to repair a bathtub that is peeling

This looks like this is one of those re-lined tubs. Get that caulked in as soon as possible. I don't mean to be an alarmist, but I am surprised that the sidewalls are set behind the tub's top edge, not over the tubs edge. The way this is now, relies only on the integrity of the caulk to keep the water out from the actual tub under the liner, where it would be trapped.

Would you be able to have the tub area redone?

If that is not possible, practically any hardware store, big or small will have a mildew resistant caulk that will take care of the immediate problem. Do remove the old caulk by carefully cutting the openings that are there bigger with out scratching the plastic tub or wall liner.

You need to buy something to break up the rust fusion you have going on. WD-40 makes a variety but there are many at your local DIY center.

I would then tap around the drain. Then get a blunt/dull metal block or chisel and hold it against the inner straps and tap on those counter-clockwise. The tapping (with a hammer or mallet) is probably the key to breaking up the rust bond. You can't just start cracking it hard or you could snap one of the center straps or warp the outside.