I doubt they are natural (i.e. not cultured). A non-graduated strand is unusual enough, finding one 'lost' in a nondescript location not likely at all. I mean, there's always a winner for the lottery, but this doesn't man that the chances to get it...
Given the size and description of the pearls, they'd be oldish akoya or freshwater cultured pearls. It
isn't too difficult to get a new strand of either for the money.
Unusually fine akoya would be more, but you are saying that the pearls do have some defects, so...
No clue about the 'H' signature.
Also, I am not sure about the guarantee: it is customary that appraisals state at least doble the 'shelf price' of jewelry. It isn't hard at all to get an appraisal for more then the price paid for th pearls. And a GIA report would cost quite a bit and take a while to obtain.
Could be wrong, of course....