There is no list as far as I know.
It's quite clear you don't know a lot about PHP - generally anything that begins with a $ sign is a variable...
It'll be quite time consuming to generate a list of variables, and into the bargain, a code change usually results in the changing of a variable - be it that variables name, operation or existence. This results it quite a lot of time spent on something that would be useless after a few changes
Check out the Documentation for phpbb3 - it lists some variables of the classes and their functions