Tile flooring question?

You can, but the real key to getting tile to stay in good shape is laying in on a clean, flat surface. It would benefit you to get the glue and old paint up if it will cause the tile piece you lay over that area will be slightly higher then the ones surrounding it. In a short time it will wobble when you step on it even after it's set with grout. Try acetone on the glue. Pour it on full strength let it sit for a few then use a sharp scrapper. I've even used a wood chisel because its very sharp and heavy duty. The paint will also come up with acetone. I had to replace 17 tiles in my kitchen when I first bought this house because the other owner left the same things down and laid tile over it. Two tiles were cracked and all the grout around those kept falling apart.

Once your mesh is screwed down and you put down the thin set start dropping your tiles. Go to Lowes or home depot and watch the course.