I did my FIC part-time. It was a few years ago now, but I don't remember it being a problem.
As I recall, I took 2 weeks leave from work and completed the rest at weekends. For the ground school, I used to go after work and start at about 6.30. It did need an understanding FIC instructor but I paid the school over the odds and he was willing to make things work. Not sure about the buddying up problem, as the FIC instructors do like to have more than one on the course - I used to have my own plane so the instructor could work me on some 'down time'.
Instructing part-time is no problem - though I should warn you that I receive about 5 CV's a week from people asking to be part-time instructors so the competition is fierce.