It takes a person with a lot of character to admit their mistakes. Guess what??! Welcome to the club! Although a lot of times I feel I know what I'm doing when it comes to home projects, sometimes, I get over-confident and seek solice in 'the club'.
Anyways, although you learned something from this particular lesson, let it not discourage you. You might already have 'learned' this but perhaps, as now, asking a few questions up front will help you in your quest to learn DIY projects. (Really, like I said, I'm no pro and I will ask questions!) Also, as an aside, sometimes, it would pay to have the pro do the prep-work and you do the finish work. Like most DIY jobs, the prep work is more important than the finish work. (The finish work is what you see; the prep work is what you DON'T see that counts just as much.)
Now, as far as trying to save the tile, it's very, very hard to do. You see, basically, you're cementing your tile to the substrate. You really DON'T want it to come loose! So, with that in mind, you may find it very difficult to remove w/o damaging it. I've never had to partake in such a job but you may be able to do it. Once, I removed a dozen or so tiles from a bathroom wall so I could install a new soap dish but it was a paint trying to clean off the backs of the tiles before I re-attached them to the wall. (Whew! I don't want to go back through that again.) As for your floor, it will be compounded 1000 fold. BUT, don't give up hope. Do check around. You may luck out. Heck, they've got stuff that will remove paint from brick these days! Maybe there's something to remove mastic / cement from tile just as easily. You never know.