Ouch! redoing the job means scraping out the old grout and replacing. In my limited experience tilers hate doing this and may miss bits, I don't think the grout is as solid afterward either, ok on walls not so good on busy floors. If the grout hasn't been sealed and given that it is so absorbent anyway it should be simple to recolour, just test first as it may change colour when it dries. Then use a quality tile and grout sealer, worth the time imho. I would hesitate to tell a professional tiler which brand of grout to use unless you have experience, they differ in quality, colour is up to you of course.

What a beautiful tile! Rule of the tump - always match the grout to the tile. Specially with such a strong patterned tile, otherwise the grout also becomes a clashing pattern. I'd pick light grey to avoid pure white cleaning issues. I have these type of tiles in my patio, you will have to be seal them with penetrating sealer after installation, but other then that you will love them forever

It's a lovely tile! I'd go light-mid grey (basically like the picture!), if you go dark then it's pointless having the interesting hexagonal shape and the veins will look a bit odd stopping and starting, and if you go too light it will compete with the veins in the marble and look to busy.

Are your floors a concrete slab or timber ? My previous home had a similar problem, but was on a concrete slab, And we had wall damage in the hall. A company like Megasealed will come and fix this in a day. They will remove the existing tiles, re water proof, and relay tiles. And also regrout the entire shower recess. If the paint in the hall is not bubbling, then you may be correct that the damage is in the floor, not the walls. And since you have not used the shower recess for a time, then it is the ideal time to get a professional in to redo the shower base. With reference to removing the hob, this may be making the job an expensive full renovation. As the hob would be built up from the floor, the tiles would not be underneath, therefore you would be having to take up that hob to the concrete/timber base, then the domino reaction may take into effect. New floor tiles, new levelling of floor etc etc.