I'm so happy for you. I was thinking of you at the weekend and wondering how you were getting on. It's been great to read this from start to finish and led me to sign up for a 10K next year. I never even dreamed I'd be able to run 5k as I have lung disease but yesterday I did it for the first time (with a few walk breaks). As long as I can have a quarter of your determination I'm sure I'll make it.
Ironically my doctor told me to run twice a day to improve my lung function. I didn't quite manage that but spent 2 months running once a day. I got shin splints and very sore knees and saw very little improvement.
It's so tempting to keep pushing but resting really works. I went on holiday and did loads of walking but no running at all. When I got back I'd massively improved.
I usually only do a minute of walking before I start jogging but I go very slow to begin with. This should also help with your breathing. I think in these early stages it's best to go at a comfortable pace, even if it's almost as slow as your walking speed. I was amazed when I slowed down how much longer I could run for.
If I'm running outside I tend to be concentrating on not going too fast and not tripping over. If on the treadmill, I usually either play music in my head or have imaginary conversations with famous people.
I think it would be nice occasionally to run with somebody but mostly I like being on my own - I usually feel more confident afterwards and always happier (unless it's been an absolute stinker).