If it is a METV, your stays in Thailand are maximum 60 days, and can extend for another 30 days. You can leave and re-enter any number of times during the 6 months that the visa is valid. You can calculate the number of times you can do re-entries.
However, the validity of the visa starts when it is issued by your embassy, NOT when you enter Thailand.
If you arrive in Thailand 1 month after it was issued, you will have 5 months left of the METV—and this will affect the number of times you will be able to do re-entries.
It depends on nationality, but OP will have to certify her diploma and transcripts at her embassy. This is needed for the school to prepare the B visa documentation.
Changing ED visa extensions for a new school might be possible, but also complicated, probably involving the permission of your previous school or things like that. Might or might not work in the end.