I have NO evidence that the NRMP is "fixed" nor do I believe it is.
Positions may open up as late as May for a number of reasons:
1) programs are not required to offer all their available residency spots in the NRMP; thus they may have others available which they can offer to anyone they like
2) programs may have an opening when a candidate breaks his contract or for some other reason cannot fulfill it (serious illness, visa or ECFMG certifiate problems). This opening can then be filled by another candidate.
3) candidates who had agreed to a position out of match may also renege and the spot opens up for someone else.
4) residencies which DON'T fill may transfer their unfilled positions to another residency track at the same institution. This actually happens fairly frequently as long as the other program feels able to handle the additional residents and the money is available for the position (which it should be - just the cost center is transferred).
As far as paying for a position, perhaps it does happen but as I'm sure you know, it is highly unethical and probably illegal. You would be wise to stay away from a PD and a program which would condone such behavior.
Meeting PDs DOES help in getting positions; this is why most residency books recommend that you do an Audition rotation - this is true for the US grad as well as an IMG. With some programs getting literally thousands of applications, an applicant who is a known entity will stand a much better chance (provided that they performed well or impressed the PD, faculty, etc.) than just a faceless name on an ERAS application. This is also the case with the scramble and any post-match contracts. Nothing fishy about it and it seems to benefit both the program and the applicant. If some IMGs are better positioned to meet PDs (ie, by being in the US during the scramble and the weeks after) tand this results in a position, then so be it - life is not always fair. There is nothing illegal or wrong with approaching a PD for a personal meeting about a position and then getting that position over someone else who may or may not be (more) qualified. People are offered positions outside of the match all the time.I'm truly sorry it didn't work out well for you Dreamer - but there is no guarantee that meeting with a PD will HELP. In some cases it may hurt (depending on the social awkwardness, personality, impenetrable accent, etc. of the candidate). Even US grads are "warned" that auditions can hurt them if they are stronger on paper than in person.
It is a matter of playing the odds, being in the righr place at the right time, early bird gets the worm, and any number of other aphorisms.
Just my 2 cents...