If I were to go to to Harbin Ice and Snow world in the morning, would I have to leave at 12 or could I stay longer?

No. Actually, you can stay in the Ice and Snow World as long as you wish, but it may be very cold.

As for the attractions, you may refer to the scenic spots for the festival, such as Zhaolin Park, Snow World and Island Park. If so, generally, it is suggested to stay 2-3 hours in each and Zhaolin Park is more worthy at night, when you will enjoy the Lantern show (formerly Ice Lantern Garden Party).