Removing Shower Tiles from Waterproof membrane
1) if you can yank the tiles without damaging the waterproofing layer below, you are fine.  You can do a water test leak once the broken tiles are removed.  Plug the drain, put a stack of coins on the floor, put water up to the coins level and check back in 24h.

2) you can regrout no problem. not sure what you call waterproof grout, but you can use any.  its purpose is not to hold water above the tiles.  The shower is designed to handle water that penetrate under the tiles and redirect it to the drain.

that said, what exactly is that black thing under your tiles?  Are you sure it is the waterproof membrane?  it might just be your deck mud.  In that case you could just patch it.

redoing a shower is a lot of work, thus make sure you have no other option first.