Does the Game Continue Even After You Have Quit the Game?

No, all processing of your sims and their world stops when you quit and starts where you left off - almost. The only thing is, sometimes your sims will no longer be in the middle of what they were doing when you left them. Sometimes the game will have made them forget what they were doing and be standing outside their house when you restart the game. Sometimes they might be in a different mood too. But all long term progress like relationships and skills will be just how you left them.