Wayne Connolly It is now 30 days for land and air arrivals into Thailand. This can be extended for a further 30 days at immigration for (I think) 1,900 Thai Baht.
Thai Immigration at Mae Sai will not let you leave the county unless you have one. You should have got one when you arrived in Thailand. It is attached to the arrival card which all foreigners must complete on arrival. It should be stapled into your passport. If you have lost it then you will need a police report and go to Immigration to obtain another.
In my opinion it looks like 7 Sep 2018 was your departure date from Bangkok, Thailand. The one in my passport looks the same but with a different departure date.
Not necessary. But of course, the school may require it. Perhaps you could negotiate with the school. Start teaching for the school based on your degree and enroll in a TEFL course. You would have little difficulty completing it.