Grout is going to need to be redone over time. Especially in those areas that take on any flex at all (like right next to the tub).

Rust would indicate metal of some kind. Either a trim piece or the tub insert itself is like cast iron (which in that case you would be able to keep it).

Looking at the photos, that seems to be very normal and expected grout wear. You can dig the grout out and re grout, and then caulk along that tub line.

Or yeah if you want to just redo the whole floor. Just get a sledge hammer and start breaking it up.

But none of that is alarming at all.