Do flights of a specific callsign always depart and arrive at the same airports?
You may be confusing callsign (with can be multiple things) and flight code/number as FlightAware may display one or both of these things depending on whats applicable. FlightAware tracks publicly available filed flights at a variety of levels depending on what callsign you search/are looking at the answer may change. For example 

American Airlines runs flight AA66 (AAL66) from JFK to BCN and the code AA66 is used to identify that flight (AA67 is the return flight). This code may also be used for a code share. Generally these codes stay fairly consistent unless there is an overlap (see ymb1's) comment for how dupes are handled.  

Generally non scheduled commercial and private traffic the flight files with their tail number, for example in the US tail numbers are often 5 (or possibly 4) digits and a preceding N. So N12345 may file for a flight from anywhere to anywhere they want to go. This history log is a good example of a charter plane that seems to fly out of KDYL to lots of places.