(Q1) I got/had my hair cut yesterday. == I had/got a haircut yesterday. They both mean the same thing. You likely went to a barber shop or a salon, and paid for a haircut :)
'I got/had my hair cut yesterday' would often be said if you wanted someone to notice (and possibly compliment) your new hair style. e.g. 'oh, by the way, I got my hair cut yesterday ... what do you think?'.
'I had/got a haircut yesterday.' would be used maybe when you're telling someone generally about what you did with your time yesterday. e.g. 'I haven't done anything lately, apart from yesterday ... I got a hair cut'.
'I cut my hair yesterday' would almost always mean 'I cut my hair yesterday (with my own hands)'.
(Q2) Yes, you can definitely say 'It's about time you GET your hair cut.?
In my opinion, this version sounds more authoritarian. Maybe something a mother would say to a son in disgust (when his hair is down to his lower back for example).
Hope that helps a little ^^