1. Lots of means this can happen. "Professional poker player" is generally taken to mean "derives primary source of income from playing poker" or sometimes "spends the majority of their time playing poker". There's no exact qualification.
2. Not generally sponsorship driven, although there are some modes of sponsorship that do factor in in some ways - the primary means of income is basically by winning money from other players. A secondary component is often rakeback, in online play. In cash games, you can join and leave at any time and your winnings or losses are simply the amount you are up or down in that particular session. In tourmanent play there is a payout structure based on your placement in the tournament (often something like 25% of the total prize pool to 1st, 15% to second, etc.)
3. It can be done part time. There is nothing to say that your bankroll can't be seeded or supplemented by external means, and you can play cash games for short or long periods. Tournaments are typically long (at least if you remain in them a long time, though you can be eliminated at any time, basically). I used to play approximately 40 hours a week (since I was treating it like a job), but now I play 10-15 and it represents about 30-40% of my yearly income.
4. I personally quit because I found the stress associated with having it as the sole means of providing myself too overwhelming. One can have a stretch of negative earnings that can last hours, days, weeks, even months - and it can be psychologically damaging in some ways. I also found that I preferred to keep it as a hobby than a means of income, since I enjoyed it more that way.
