Is peel and stick vinyl tile flooring really waterproof?
They are waterproof in that you can get the top as wet as you want.  They are not waterproof in that they will keep water off your floor, nor will said water not affect the bond to the floor.  These tiles may be able to last sitting in water but they will not keep water from getting underneath.  Your only options are really one sheet of vinyl or tile (could I guess use a membrane).  

If you install these right though I would bet that 99% of the water (given you don't just let an inch of standing water sit) will never meet your subfloor.  No caulk is needed, just make sure that all of the edges are tight and clicked into place.  To keep your bathroom better suited for a kid you may want to make sure baseboard are sitting right on these and caulk between baseboard and tile.