"I have an M.A"
"I am a graduate" YES.
Do we also say "I am a masters"? NO
Question3- Should that be masters (plural) or master (singular)?
It is not a plural noun - it is the possessive 's'
It is often said as, "I have a master's from such and such a University",
where master's is short for 'a master's degree from...'
- For Teachers