Help! Tile falling off wall around bathtub/shower

I am not much more than a novice in plumbing matters but when our downstairs bathtub recently began draining rather slowly I thought snaking the tub was (or is) the remedy if there were no obvious causes (hair or some foreign object blocking the drain that could be removed by hand). Actually I did remove a number of raisins compliments of my 3 yr. old boy and some of my wife's shoulder length hair. However, the tub still drains too slow so I bought a snake w/the intention of further clearing the drain pipe. The instructions direct the user to remove the overflow plate and work the pipe through the opening in the tub (I hope I said that right, the instructions are in the closet where the raisin fiend is sleeping right now). Anyway, that was not a difficult task. At this point I noticed that even when the pipe was connected before I had removed the overflow plate, the pipe was not flush with the tub (fairly big gap between the tub and the pipe). The house is on a slab and it appeared that any excess water from the tub was just running down the outside of the tub and falling to a cut-out area of the slab (filled w/stones). Also, when someone turned on the sink in the same bathroom, water would come up through the tub drain. Is the gap between the overflow pipe and cover something I should try to remedy or leave it to a professional? If I can fix it, should I proceed to snake from that area? Any help would be appreciated.

We have a problem with toilet sweat. Water collects on the outside of the tank and drips off. It looks bad, but more than that it's a nuisance and causes black grout to build around the toilet. We have one of those trays under the tank to catch the moisture, but it is not helping. Still ugly and still dripping on the floor. Instead of trying to live with it, is there something to actually prevent this toilet sweat? thanks