Toilet not flushing properly because flapper is closing too soon
Sounds like the chain is too short. The flapper has to come up past vertical, so that it's out of the way for the full flush. Once the tank starts to fill again, the flapper will slam shut.  

If the chain is too long, it won't lift the flapper enough. If the chain is too short, it won't let the flapper fall into the full open position. The chain has to be just right, for everything to work.