I am no expert, but it appears (from the wood, shape and handles) to be a danish teak dresser - they date from around the mid-late fifties and sixties. Don't think they are particularly rare (check for makers mark eg) Borge Mogensen/Sorborg Mobler?), but they are lovely, and very on-trend (similar new version by West Elm sells for $1099). Didn't find one exactly like yours on ebay, but similar vintage ones listed start at around AU $550-$600 and go up from there. A great thrifting find!

Perhaps you could post some photos of your selections? I am having trouble visualising what a blue marble-look floor tile actually is? Could you use it as a splashback tile?

Have you priced having the whole lot re waterproofed and tiled? You are already doing the hardest section, it may not be that much more to do the lot. Otherwise agree with above match to the white tiles in the rest of the bathroom. But it will always look like a repair.

this is tricky but it can be done! even with darker floor and vanity colours, chosing large and lighter coloured tiles is important. i think if you do go with the slate colour, keeping taps etc in a stylish black colour will complement those changes. if you do go with the marble, choose brushed nickel because the black will be too stark of a contrast. just my two cents, of course :)