My suggestion is first talk to the builder to see what instructions he gave the tiler as i'm sure he doesn't want a dissatisfied client. Most tilers will check with the builder before grouting especially when a colour is not specified in a contract. I would then ask if the grout could be replaced, from the looks of it it will only be a couple of hours work. There is nothing worse than having to look at something you are not happy with. I would not use the paint product as it will alway look like it is covering a mistake. White will look better. Make sure the mixer body is waterproofed or you will have a leak in your wall, the membrane should lap onto the body from the wall if it is done well. All the best.

I would guess it is some kind of expansion joint. To allow for expansion and contraction of subfloor.

Timber floorboards may compete with your wonderful ceiling. Terrazzo may be too busy and also compete against your wonderful ceiling. Concrete look MATT, rectified edge, porcelain tiles would be my choice out of the three options. They are a natural warm texture and would look great against a white kitchen and the timber ceiling.

In an apartment you need to check before you add floating floors. Some buildings have regulations to assist with noise control. If you keep the tiles, do they have the same under tones as the paint you’re considering using?- a clash between a cool and warm white will always look odd.