As for stool pigeon, the etymology is a bit more complicated. Today, we think of a stool pigeon (or stoolie) as an informer...but when the phrase first appeared in the United States some 170 years ago, it generally meant a person who was used as a decoy to entice criminals into a trap.
This usuage has a long history. While most contemporary accounts note that the term comes from the practice in hunting of tying or nailing a dead pigeon to a stool to act as a decoy, that's probably a bit of a stretch. The most puzzling aspect is why this particular bird and why this particular piece of furniture? Another approach is to follow the trail of the word "decoy" rather than the described act. One form of "decoy" is the archaic term stale, which , in turn, may well have evolved from the French estale, in which a pigeon was used to entice a hawk into a net. Sound like we're getting closer? The term shows up in English in the early 1400's and within the next hundred years was being used not only to describe such a hunting lure, but also to describe an individual who entrapped another person. By this time it was also beginning to be applied specifically to a pickpocket's accomplice. [By the way, we know this variation in another form today -- as the verb "stall" as in "stalling for time" which is what the pickpocket's assistant does to give the thief a chance to apply his/her trade.]
Inasmuch as "pigeon" has been a term applied to a fool for almost five centuries, one might ask why not the Three Pigeons rather than the Three Stooges? Perhaps there's a link in the theatrical connection to the term "stooge" which can refer to the partner in a comedy team who feeds lines to the other comedian. Remember the original name was "Ted Healy and His Stooges."