This is a somewhat painful subject for me because I feel that I didn't handle this too well.
I moved away for similar reasons you mentioned. I wanted to distance myself from everything and everyone so I could discover myself more freely, without complicating the lives of others, at least that's how I rationalized it to myself.
But in reality there was another important aspect, I did not want to look into the eyes of my friends and family while doing it(even after coming out to my parents), I felt ashamed of myself.
1) How many months after starting HRT did you do this move? Were you passing well already when you moved?
Before I left I got the psychologist and psychiatrist letters required to go on HRT but did not start HRT. I was passing without HRT, my two problems were facial hair and voice.
2) Upon completing the move, did you immediately start getting involved in your new community after moving, or did you wait until you thought you passed almost flawlessly?
I got more involved in the trans community almost immediately, I helped our local trans organization to organize quite a few events and conferences.
Otherwise I mostly kept to myself and this was not hard to do in the middle of a big city where nobody really cares about anybody. There was no community to speak of in a city like that, I barely spoke a few words with my neighbors in years.
I could not afford to not work until I felt ready to go full-time so I started to lead a double life, go to work as male, almost everywhere else as female.
3) How long did it take to build a new network of friends? How did you make these friends?
It really depends on how we define friends but I think after 4-6 months I had at least a few people to talk to.
I made friends at my day job and at the events organized by the trans organization I mentioned above. I had two circles of friends and they were strictly separated.
4) Did you maintain contact with supportive friends from your old life?
I basically abandoned the few friends I had without much explanation because I was too weak at the time to deal with the situation.
I started writing multiple letters to my friends to explain the situation but as time passed I felt worse and worse about it and I never sent any of them. In the end they did not have a chance to accept or reject me. This is something that I regret to this day.
It was a complicated time because I had to find a job to avoid starving(literally), then after I got a job I started to build a new career which took a lot of time and effort. I moved forward in my life a great deal since I moved here, but I'm still not legally female, so still leading a double life. The facial hair is mostly gone and I'm slowly making some progress with my voice as well.
I have a few co-workers who I consider a friend and I'm hoping they will at least try to understand my situation when I come out to them.