Have you tried a different contractor?

Not yet. A little skittish as this one was a recommendation from someone I trusted. :/

Something's not translating correctly. :) Of course you can remove an old tile installation! But it's a dirty, filthy, job, and is sure to have repair issues with the subfloor underneath it before you can do another tile job on top. As far as applying another floor on top of it, yes that will create height issues, including needing to undercut your baseboards and needing transitions where the taller floor meets a lower floor.

Perhaps you need to have a good sit down conversation with your installer to be sure that you are both on the same page. I suspect there is some communication difficulty impeding understanding between you.

I think we can chalk some of this up to "I don't want to do it for the price stated". Any tile floor can be removed at any time...the cost to do so is quite high. The time to do it is quite long. The dust produced is quite a bit. I know many, many contractors (wood flooring installers) who won't go anywhere near a "tile rip out" job because it is so hard on their bodies and they don't make a lot of money off of the demolition. It is the installation they make money off of.

I would agree that painting it probably won't work...very well.. Carpet can be done...but you will need to "fill in" the grout lines (read: sand tiles and then add thin-set/cement leveling compound) prior to installation. Floor heights are dealt with ALL THE TIME. Transition strips, uneven T-Mouldings, Reducers, etc have been used for "step downs" as much as 3/4"!!! No problem.

Personally I think he would like to move on from your job to another client.

Thank you, everyone for the feedback. It is much appreciated! - BTW: the tile was cemented on top of the concrete foundation. We know this because of the work he did with the gray. - Thank you, Cancork Floor, for the advice. At least now we can move forward with more confidence. :)