As far as the flight ticket goes, I have always bought a 12-month return flight. But, once you have a resident permit in your passport, you can leave and return to China as often as you like. You just need a return ticket for the first time.
When you get your visa, it will be valid for entry to China for three months, and it will come into operation on the day you enter the country. I do not think that the date on the invitation letter has a lot of relevance, except for when they need you at the school, for orientation, and to start the process of getting the Z-Visa changed to a Residence Permit.
I would be hesitant about buying the air tickets, until you have the Z-Visa in your passport. But maybe that is just me, being cautious of any (unlikely) delays with the visa processing.