Nothing to worry about, M&P 1.0s would automatically chamber a round when a magazine was firmly inserted, and up until the 2.0 came along and corrected it, most folks just considered it a feature which was entirely beneficial in a self-defense scenario since it saved time from having to hit the slide release or slingshot the slide.
As for accuracy, it's tough to tell until you've shot it more, but a lot of folks report poor accuracy on their first range trip after buying a Sigma Series pistol due to the long, heavy trigger pull, but once they've gotten used to it they find that the gun itself is plenty accurate.
The best advice I can give is to treat the trigger as if you're shooting a revolver and let it roll straight through when you pull it. Don't try to stage it or squeeze it really slowly in attempt to keep it super steady on target, just roll right on through it like you're shooting a revolver. Don't worry too much about your accuracy at first, your goal is to get a feel for the trigger. In fact, you could dry fire it, (use snap caps if you've got any) but firing live ammo is best, that way you can evaluate whether your accuracy is improving as you grow accustomed to the trigger pull. As long as there's nothing physically wrong with the pistol, accuracy ought to improve as you continue to shoot the gun and get a feel for it.
Good luck.
