We renovated recently and removed an old cast iron tub that had been reglazed badly several times - it was tough for our plumbing team to find someone to cut it up and get it down the narrow stairs.. we opted for an acrylic model for our replacement, I absolutely love old things too but definitely appreciate the new pipes that our team was able to run to the new bath as well as a nice level subfloor 😅

You want to pull the vanity all the way to the left? Here is what we did when remodeling a small guest bath in our Florida home. It came with a vanity wall to wall. We replaced with a smaller vanity with a left offset sink. Now there is room on the right to hind the toilet bowl stand, room on the left for trash can. That bottom drawer holds so many hand towels. The drawers make more sense too than just doors. It appears you have a similar problem having in front of the toilet to wash hands.

Tub to Shower remodel

As in being billed for him buying one or renting one? Or does he have a line item on an invoice for the usage of one? It all depends on the agreement you made with the tile installer and the wording in the contract. But to me it would not be a good sign that a tile installer did not already own one.