Should I just finish this year and then come back in September like it's my first year and sort it out at that time?
That depends on whether you're an EU citizen or not. If you are, just leave it: there's pretty much nothing which
must be sorted out. If you aren't, you might want to plan around not coming back as you have almost certainly already over-stayed your visa.
one of the first things the employer did was say "You must have PESEL"
They are wrong, you do not need a PESEL number (their internal procedures may require a PESEL number but in that case you just make one up).
Some sources tell me if I buy a car I'll need a PESEL to register it or get the insurance.
I have a car (which was recently re-registered) and it has insurance: I do not have a PESEL number.
If nothing comes, it might be better (though illegal) to draw a line under the matter when making future tax declarations rather than opening a can of worms.
It depends on if he/she is an EU citizen or not and on what basis he/she has been employed. Personally, if I was an EU citizen who'd been employed on an umowa o dzielo basis and the school refused to give me copies of the paperwork showing that I'd been paying tax, I'd probably pass that information on to the tax office myself.