No, there's no need to remove the tiles unless they're loose or chipping all over the place.
If the vast majority of the tile is firmly glued to the floor, leave it, and fix the bad spots. Of course, you can use any tile that's the same thickness/size. Scrape the old glue out of the area being fixed and put new glue in that's used for concrete & tile. IF the tiles separate from the floor by simply walking on them, then if you were to simply cover them up with pad and carpet, you'll have a problem of 'movement' underneath. In this case, get an ice chopper with a nice, sharp blade (use a metal file) and start scraping them up. After you've gotten them up, (and keeping that ice chopper sharp) knock down the 'high spots' before you have carpeting installed.
With all this, assuming you DON'T have a water/moisture problem, you don't need a moisture barrier installed. The tiles don't provide any insulation nor are they considered a moisture barrier. If there's a spot on the floor that NEVER had any glued tiles, get a sheet of 3' x 3' 6 mil plastic and duct-tape all 4 sides COMPLETELY to the floor. If, in a few days, you don't see any condensation on the underside of the plastic, you don't have a moisture problem. What you might want to consider doing is asking your carpet retailer/installer if a moisture barrier should be put down first. If so, they may suggest a sub-floor since you may hear the plastic moving under the padding + carpeting when you walk on it.