Yes, picar means also bite (when a bug does it).
The meanings of picar from the top of my head:
-Sting -> Me picó un mosquito.
-Itch -> Me pica el brazo (as in, it's itching).
-Take the bait -> Ha picado un pez / Has picado (referring to a prank)
-Annoy, provoke -> This is like... someone will get easily involved in a competition, soy you do that to them -> Me has picado, ahora quiero ganar.
-Get annoyed -> Siempre se pica por tonterías.
There's a lot of meanings, haha... The only thing you can do to tell them apart is context, I think. An arm won't sting anyone, right?