How can I remove the stench from a p-trap?
Safety point: Never mix chlorine and other chemicals unless they're intended to be mixed with it. If you want to try a chlorine pill -- or chlorine laundry bleach, which is essentially the same thing -- run lots of clear water through it first to rinse it out, and do the same after bleaching.

Yes, a clogged vent may force sewer gasses to escape by bubbling through the trap. Typically, drain vents look like a pipe sticking straight up out of the roof for a foot or more.