She definitely would have gone through the proper (Ministry-approved) process of becoming an Animagus. I believe there would be some paperwork involved in that, and a lot of training. It probably took her a couple of years. Also, she might have written essays for "Transfiguration Today" and done research in the field.
A: So, we think she was born 4 October 1925. In Phoenix, she says that she held a position at Hogwarts 'thirty-nine years this December' (321). That would be in 1995, so she was appointed in 1956, when she was thirty. What did she do before then? What skills would she have acquired?
By close relationships, do you mean that in a romantic or in a friendship-sort of sense?
B: Most people she McGonagall as this 'stick in the mud' character, but do you suppose she was involved with other people? (I don't think she ever married, I'm just saying there were close relationships with people.) What happened?
Romantic - I can see her wanting to focus on her career in Transfiguration (maybe she had the set goal of becoming a teacher?), and a relationship failing because of that. Maybe she didn't even have that strong a romantic interest in anyone. Or maybe she did, and was in love, and they died during the war.
Friendship - she'd still be friends with them. She's very close to Dumbledore. I don't see them involved romantically in any way (not even unrequited love), but they are close friends. She seems to get along very well with the rest of the staff too.
I'm quite sure that he hired her. I don't know if Dippet died or retired... if he retired, he might have offered Dumbledore the job months before the actual event. Then Dumbledore would have owled McGonagall about this, as they possibly still had contact... I can see her as a sort of favourite student of his. She probably was very talented, especially in Transfiguration, and he might have known that she was studying in this field even after school. I can see him directly asking her to take his old post.
C: If Dumbledore was Transfiguration professor in 1943 (CoS), then did he leave before he became Headmaster. There must have been someone who acted as Transfiguration professor between him and McGonagall. Depending on when he took the Headmaster post, dd he interview McGonagall for the Transfiguration position? Who did?
This might be a bit cliché, but I can see her as sort of a Hermione-ish character. She would have been studious (though probably mostly in Transfiguration, I think she would have had subjects she wasn't too interested in), but she was in Gryffindor, and I think that streak would have shown in her. I can definitely see her on night-time strolls during the castle. I'm not saying she was constantly up to mischief or anything like that, but I don't think she'd have stuck to the rules too much either (when she didn't see need for it).
D: Going on this 'stick in the mud' thing, what are the chances McGonagall did interesting stuff when she was younger? Not crazy things, but let's say she wasn't always acting like the proper old lady? (I don't even want to phrase it that way...).
I don't think it took an error to make her that way. I think it's just the way she is. She's strict and 'uptight' around students - but she's a teacher, and don't forget what era she grew up in. Maybe she came from a strict family... She doesn't like to open up towards students too much and believes in discipline, but she can be warm and open towards friends.
E: Perhaps she made an errour when she was younger. Nothing too outlandish, but something that made her uptight. She acts a professor with a strict demeanour, but does she drop that a little when she's around close friends? She's still 'proper', yeah, but she opens up with a softer side.