My guess would be that cucumbers are an astringent. Have you ever read or seen where cucumbers were used over puffy eyes to reduce the puffiness? Same thing, I'd think, but I have no idea why it works only on mosquito bites.
A lot of times during summer I blend cucumber slices with aloe and keep the thick results in the fridge. I use the combo to rub on insect bites and that works for me. Of course it doesn't last long, so I don't make much at a time.
Anyway, that's all I know, cucumbers cause the pores to constrict, to sort of draw together, so I guess that's what happens to the swelling from a mosquito bite. I've never used it that way, but always use it with aloe.
Ha! I'm no help at all, but maybe somebody else knows the answer regarding other bites.
Oh, and me too about eating very much hot stuff all at once. I can never get enough ice cold water!