Im guessing that it effects crimes, but what else does it effect? Does it effect how well you train in the Gym? How well you do in pvp battles?

Im just trying to see if its worth purchasing Wooden Chair at the Wardens office for +1 STM..

Yes, the higher your overall stamina the better you train at the gym. For everything else, the more stamina % you have, the more successful you are. If you fight someone at 50% stamina, your stats are basically at 50% as well