How to repair a loose marble door sill

If this is real marble, you are unfortunately stuck with what you got. once a stain is set in unsealed marble, it is next to impossible to remove. The only method is to mechanically repolish the surface, removing material until you get below the stain. Marble is a very porous material. Sorry,,,,,,,

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.