It certainly appears to be a tiled base, possibly over a vinyl or lead pan, that was not done properly.
My reason for saying that is because it appears that the wall tile came down before the base tile was done. The wall tile should always overlap the floor tile to make leaking less of a possibility.
Notice I did not say that it would prevent it.
If the pan and drain have been done properly, this thing may never leak, but why force the issue. There's going to be a build up of water under that floor tile, for sure!
As far as the cracks go I would leave the grout the way it is, and go over it with Lexel clear caulking. The grout will be held in place as long as the caulk joint is wide enough to bridge it.
Trying to remove what is there will be a horror show waiting to happen, but at least the Lexel can bridge it, and be repaired with itself if necessary.
Silicone caulk can't do that after it has been applied. Any and all residue must be scrupulously cleaned or the new silicone caulk will not adhere.
Lexel, which is not silicone, does not have this problem, and it can be applied to wet surfaces if need be.
Just don't smoke around the shower while you're doing it.............
HE