"Il" does not refer to a "him" here. The active verb is "rester", which is "to remain". "Il" should be treated as part of an expression, as it represents a situation, similarly to "il fait froid", or "il est dificile de parler"..
"He doesn't have any more tickets" would translate to simply "il n'a plus de billets".
The essential meaning is that someone has run out of tickets to hand out. But from the French sentence on hand, it cannot be inferred whether that person is a male, a female, or the box office.
"Il" in this case isn't a person. It's the same "il" from "il y a," "il faut," "il neige," etc. So "Il ne reste plus," to my understanding, just means "There are no more," without talking about any individual. I'm not quite sure how the sentence you mentioned would be translated, though.
I would say yes, just considering the meaning. If you say there's nothing left of something then you'll always be referring to a quantity in the plural, so you'd say "il ne reste plus de chaussures" (there are no more shoes left) "il ne reste plus de bananes" (there are no more bananas left) and so on ...