About 8 months ago I removed what I believe is exactly as you've described: a metal framed sliding glass door from a cast iron bath tub with tile up the walls. (I then promptly destroyed the tile and tub, but that's irrelevant to this discussion...) There had been some holes drilled in the wall tiles, but the tub itself seemed to be fully intact. There was a lot of caulk everywhere that would need to be scraped off and then cleaned up with some sort of chemical. (I found a caulk remover from one of the big box stores that was fairly cheap, easy to use, and didn't do any damage to any acrylic tub so certainly cast iron and tile could handle it.) I'm guessing you also may end up with some holes in your tile where the door is secured that you may need to patch and/or cover with the new door.

To avoid scratching it, do not use any abrasive materials. Try acetone in a hidden area. It may soften the paint enough to remove it with a pliable plastic scraper. Michael

I have painted several steel doors and never removed the moulding and glass. Why ask for problems? A steady hand keeps the paint where it is supposed to go, and a quick rag or your clean finger removes any that may go astray.

You would need to remove the exterior glazing beads in order to remove the glass. The glazing beads degrade with age and could very likely break. You can get new ones from Andersen, if need be. The plastic plugs on the interior of the sash are for interior grilles to fit onto the sash.