Such as from handling hot peppers or Bengay, Icy Hot types of creams. What exactly is that sensation? What is stimulating what type of nerve endings? Ditto for the cooling sensation of mentholated stuff and muscle pain creams.
Bonus question: why do hot peppers burn my mouth for only 5 minutes but my skin for hours?
On your bonus question, I think a lot of the answer has to do with saliva. Besides the wet properties, it does contain some enzymes to start the digestion process. I suspect that if you continually licked your regular skin, you’d see much of the same benefit.
The active ingredient in hot peppers, and in some of the medications you describe, is capsicum. Essentially, the burning you feel is a chemical reaction.
This may not have been exactly the answer you were looking for, but I’m not a biologist.
That would make sense, that the sensation is more of an irritation than anything else. Still wondering about muscle pain creams, though.
As for licking myself, well. 
Two types of nerve cells (TRP-M8 and TRP-V1**) **allow us to feel the sensations of cold and heat. A protein causes these cells to react and absorb an extra sodium ion, which triggers an electrical current, which tells the brain that heat or cold is being felt. Normally this protein would be released due to exposure to actual heat or cold, but some molecules like capsicum and menthol trigger the same reaction by binding to the right neuroreceptors. That is a simplified version but it’s the basic idea.
Interesting, thanks. So it is the same nerve cells that register actual heat or cold that are “misfiring”.
Here is a good description of the receptors of heat, cold, and pain: http://users.rcn.com/jkimball.ma.ultranet/BiologyPages/P/Pain.html
BTW, the chemical in peppers is capsaicin. Capsicum is the name for the peppers themselves.