You need a 3/4" layer of plywood, followed by a layer of hardi-backer or cement board, before you thinset and tile a floor.Your subfloor ideally should be level, and at the very least, flat.Without meeting these conditions your tiles will crack.

First a couple of questions, 1, What is on the other side of the cement wall?

I just went through this process and my firewall was approved by they city, but only after I replaced all screws with cement coated fire-rated nails. 

But if you have an RV or equipment trailer you are storing on anything but a cement pad or blacktop you get a written warning.

Should these layers of adhesive vinyl be removed before installing the new cement board and ceramic tiles?

You only get close to $600 for vinyl shakes and fiber cement or wood painted how ere detailed trim can add to the cost especially for Azek or equivalent.I would check your numbers.

In each bathroom there is what appears to be an older clawfoot tub that has been "built-in" with painted plywood or cement board. 

A couple of day laborers, a few posts, and a couple bags of cement and you can have it up in a few hours. 

No cement.

It was frozen to the cement so I couldn't lift it to look underneath...