The breaker in the panel is a 15A with 14-2 wire leading out to the junction box, at the junction box all (3) other wires leading out are 12-2. (They go to wall switches & lights, running computers, tv's and stuff like that.) Is there any problem with this current wiring?

14-2 is the lightest wire you can use with a 15a breaker, you can use 14 or anything heavier if you wish. Just make sure no one has installed 20a receotacles down the line because they noticed the 12-2 wire, got to stay 15a