I have run into the same situation many times. Let me say first, that I don't condone doing work that requires permits without them, as it often leads to crappy quality or safety issues. I always prefer to do quality work, but occasionally budgets make it necessary to do a quicky fix.
In your case, the most important thing is to strip away all old wax and dirt on the existing tiles. Tack down any loose tiles, especially around the edges with ring nails or a power stapler.
There are many de-waxing products and can be found at any box or hardware store. An alternative cleaner is ammonia and water. Cheap and effective, but smelly. You can also use a mix of regular bleach, TSP, and water. (BTW, never mix bleach and ammonia!!!!) Clean the existing tile with one of these products and scrub the surface well with a mesh type pad, like a 3M green scrubbie. Be sure to rinse the area well with clean water and allow to dry.
You may also consider a tile primer. This product is a bit pricey at over $30 a gal, but when applied over most any hard surface gives an excellent base for any self adhesive tile product.
When you get to the bathroom, remove the toilet and cut your tile around the closet flange, then reinstall the toilet. It is only two bolts, one water connection, and a replacement wax ring. It will look 100% better and avoid tiles coming up due to the toilet sweating and wetting the edges of tiles cut around the base. Only takes a few extra minutes, but will make a big difference.
What kind of flooring are you putting down? If it's tile, I'd probably reinforce the floor a bit first with a new layer of t&g subfloor before putting the tile down... which solves your problem.
If there's any musty smell, you can clean the floor with trisodium phosphate (the real stuff, not "substitute") ... but don't soak it. Just scrub it with the tsp and a scrub brush. The TSP will probably take up the glue as well... but has a significant chance of eating your plywood in the process. ;)
Best Answer
I have seen self stick tile not lasting as long as it should. Tile with adhesive is the way to go if you want durability.