Is it possible to use waterproof grout + sealer to fix shower room floor leaks?
You can remove the existing grout, and re-grout with an epoxy grout. It's more expensive than regular grout but it's completely waterproof and you won't need too seal it. It's also much stronger than regular grout. 

However, if in fact the shower pan is broken, then this will probably be a temporary fix and you'll know when it fails because the leak will start again. A permanent solution would be to rip up the tile, replace or repair the pan and re-tile and grout. good luck