Neutered Male Cat Behaving Like a Cat in Heat

Is it possible my neutered male cat is a hermaphrodite? The way we knew it was time to get the other male cats in the house fixed was when they tried to mate with him (both before and after he was neutered). Also, we had four cats neutered at the same time, and it all went normally, but when the vet brought him out, she said she thought he might die, at least, that's what my husband thinks. We live in China, so he's not sure what the vet was saying. It'd be awfully nice if we could ask her if anything was weird in there, but she wants $350 because one of our cats scratched her (I don't even remember the incident, she just informed my neighbor who helped me later, and besides, wouldn't you think that'd be a hazard of the job?), so we go to a different vet now. 

Anyway, this cat goes through phases where he just stands at the door and cries, or walks the house and yowls. I wouldn't say quite as bad as it was the single time the female cat we have went into heat before she was spayed, but it's something. None of the other cats do it. He also tries to escape regularly, mostly because he wants to rub his scent in the hallway (we live in an apartment building). The other cats occasionally target him to kind of try to mate with him, but they're neutered so it's really just like they have an instinct they don't understand. The biggest problem is that he pees on beds and nothing we do makes a difference. It has only just occurred to me that the only other cat that ever peed on a bed was the female when she was in heat. 

Is it possible he also gives off female hormones and goes through some kind of heat cycle? I mean, if that were the case, and I had any idea how to talk to my Chinese vet about it, and he could get his female parts removed, and that would end his bad behaviors, I'd be so happy.

I think that this is highly unlikely. Sometimes cats that are "fixed" will simulate mating and that is normal.  In our household my older cat uses this as a sign of his dominance.  As far as being vocal it could be just his personality however the fact that he is peeing in inappropriate places makes me wonder if he is dealing with health issues.  I would take him to your new vet and have him checked for urinary tract infection or crystals.

All of my neutered males will try to scruff and mount each other or the spayed females from time to time. It's a play/dominance thing. Some cats are just a lot more vocal than others and will yowl for no apparent reason. Does he stop if you distract him with a toy? I usually try to get them fixated on a wand toy if I want a bit of peace and quiet.

As @Miss Mew  said, the most likely reason for peeing in places he shouldn't is a UTI. Ask your new vet to check for this. You can get good enzyme cleaners on line which will get rid of the smell of cat pee so he won't think he's supposed to pee on your bed from now on. I just got one called Fizzion, it works much better than any of the others I've tried.