I've been told I can't buy a car and register and insure it unless I have a PESEL. Assuming this is true, I need a PESEL.
As far as I know (from personal experience and from the experiences of others, the most recent of which was the week before last) you do not need a PESEL number in order to register a car in Poland. What you need is a maldunek (a registered address) and your registration can only be valid for as long as your maldunek is valid for. This means that you need a Residency Registration Certificate (those are good for five years) if you want to register the car for more than three months (as a maldunek can only be valid for three months unless you have a Residency Registration Certificate). But to get your Residency Registration Certificate, you first have to have a three-month maldunek. You use your three-month maldunek to get your Residency Registration Certificate, then you use your Residency Registration Certificate to get a five-year maldunek and then you use your five-year maldunek to register your car for five years. After five years you get to repeat the whole thing all over again.
If you register yourself in someone else's flat you have to provide some sort of rental agreement. In my case, my girlfriend rents a room for a token sum a month, and I have to pay the tax on that to Urzad Skarbowy every month (17zl).
You need to be careful with that. You are clearly renting that room at significantly below the market rate for a room in Warsaw (I'd say it has got to be at least 500zl) and the tax office are perfectly entitled to assess the tax based on what they think you should be paying (and nail you for back tax too). The way to get round it (again I'm speaking from personal experience here) is to register your girlfriend as your 'konkubina'. Yes it does work.
BTW, if at all possible, it is far better to not get a PESEL number if one intends to drive in Poland. Under the Polish system, points for traffic offences are stored under your PESEL number, not under your driving licence number: no PESEL = no points.