I feel for your situation--I've known several people who have been in the same boat. I can't blame PI's for being interested in keeping their students on, as you've been fully trained now and are offering the most ROI they can expect at this point. But to answer your questions:
1) At the institutions I've been at, I do know of a few people who have delayed their graduation and stayed on AS STUDENTS after they've defended. There can be advantages to this--e.g. clock for K eligibility doesn't start until you formally graduate However, if you've already defended and the paperwork for finishing is going through the university system, I find it surprising that your PI wouldn't have to re-hire you as a postdoc. In the majority of cases where my peers have stayed on, they've been re-hired.
2)I agree with your assessment. Your PI seems to be looking out for his/her interests and not yours. If I were in your shoes and had a postdoc lined up already, I would start it immediately. The only situation where I would extend time in your current lab is if I were completing the last experiment or two on a revised manuscript to ensure its publication. Based on what you've said, there is no guarantee that you'll get anything out of staying. Just make every effort to make the transition to your new position as smooth and polite as possible. If your PI wants to go scorched earth, there's nothing you can do, just try to not exacerbate the situation and make the most out of your next job.
3)Your career is in your hands. Always try and make the best out of whatever situation you're in while trying to maintain good work/life balance. Working on cruise control or burning bridges never helps the situation.
RE: the K99 application--4 years of eligibility is tough. The time goes by really quickly. You will be better served spending 6 months in your postdoctoral position learning the ropes and gathering preliminary data towards your future work than trying to get another publication out of your PhD lab.