How to repair a marble floor corroded by acid

When I've done grout repair I've cleaned out the old grout, did some slight prodding to see if anything else was loose, made sure the area was bone dry and then got a small batch of grout from the local hardware store. You can probably buy a small tub of the premixed grout (unless you have colored grout then you're stuck buying a larger tub usually).

I just put it on as normal grout, slap it on, push it in, smooth it out a bit, let it dry and wipe off the excess. It'll take a little while to fade in but now it looks like it was part of the original tile job.

This is a tough one. There is no easy way to remedy your problem. The best way would be to remove and replace the tiles properly. If in fact the tiles are real marble, you can grind them to create the proper angle and slowly refinish the surface by stepping down grits and finally getting to a wet polish grit of 2400. That is actually impracticable for a DIY if you don't have the equipment and materials to do it. If it were me, I'd remove a couple of tiles and reset them to fix your drainage problem. A couple of hacks with a hammer, scrape off some of the thinset, replace the tiles with a little premix thinset, regrout, wait a couple of days, seal the grout and your done. The actual labor time may be only a couple of hours, even though you have to wait a bit to use the shower again.