Aspiringdoc is right - G0 is definitely not the name of the checkpoint, rather the 'resting' phase where cells aren't dividing or preparing to divide. I think when cells 'fail' the checkpoint, they probably go to apoptosis rather than into G0, but my knowledge is shaky here 
Just going to clarify G0. There are plenty of cells in the human body that don't divide. These cells are said to have entered a state of senescence, which in the context of the cell cycle is represented by G0.
It is possible for a cell to exit G0 and then proceed through the cell cycle; however, a cell in G0 is most certainly not preparing to divide.
If a cell fails the checkpoint, a number of things can happen. Typically, various enzymes will try to correct whatever has caused the cell to fail. If they can't do that, the cell will normally undergo apoptosis. It can, however, go into G0 instead.
Do the MHC actually get recognised as self or non self? Don't they just present antigens
They do present antigens. To activate a T-cell, however, it must bind to the MHC molecule AND the antigen simultaneously.