It is NOT normal for thinset to be wet, unless you mean it was wet only on the surface. It IS, however, normal (which is this product's downfall) for mastic to be wet and soft if used in a wet location (doesn't get any more wet than a shower
When you replace it, make sure you use thinset(that comes in a bag and you have to mix yourself) and not mastic(which is probably what the "experts" at home depot or lowes will suggest).
Grout with sanded grout if the spacing is more than 3/16", unsanded if less. Caulk at any change in plane (like Ron6519 said).
EDIT: Read up on tiling a shower also. It's alot different than just laying a floor. Schluter makes kits to take the place of the liner, which will make it cost more, but easier. You'll need a 2 part drain and.....well anyway, read up on it, lol. If you have any questions, come on back and someone will help ya out.