Taking your questions one at a time:
Is it acceptable to not have enough of one per person at a party and still serve the item? For instance, should you make sure that there is one cupcake per child at a birthday party , even if there is also cookies and ice cream for dessert?
I'd have one person. So at least one cupcake per child. In part, my reasoning is that kids don't always have the best manners. At a party, hyped up with the activity, the sugar, the games, it would be fairly easy for a child to melt down over the lack of a cupcake. I'd want to avoid the melt down. Also, cake is pretty traditional at a child's birthday party--if there were cupcakes instead of one big cake, I'd expect each child to get a cupcake. There would also be enough ice cream for everyone to get the same amount, and enough cookies for everyone to have at least two.
Should you prepare cupcakes for the adults that are in attendance as well?
This one, I'm kind of mixed on. Some people would say that not having cupcakes for the adults is creating an A list/B list situation, where some guests are treated better than others. In this case, "better" meaning some have cake, some don't. My own take on this is slightly different, and even more so because this is a child's birthday party. The adults are there because they are accompanying their children more than because they are invited guests. If the adults are being fed, I don't think they absolutely have to have the exact same meal as the kids, but they do need a similar meal.
So if the kids are getting dessert, the cupcakes, then then adults need a similar dessert. I might, at that point, be passing around my Navy SIL's rum balls, because all the adults might be needing a little "something" at that point. But most likely, I'd have enough cupcakes to go around. But if I came up a bit short, because a kid dropped a cupcake and needed a replacement, I'd expect the adults to understand that and not make a fuss if one of them didn't get a cupcake.
Is it ok to serve something were you only have 15 of the item but 30 guests if there isn't another equivalent? (Is ok to serve cookies and say have 15 chocolate chip nothing else instead of say 15 chocolate chip and 15 sugar cookies for 30 guests?)
So you have 30 guests and only 15 cookies total? I would probably not serve the cookies. A lot of people take two cookies at a time, so you'd have fewer than 15 people getting cookies. If the cookies were the only option for dessert, I'd just not have dessert in that case. If there were several other options for dessert, say 3 other kinds of cookies, then it wouldn't be so bad, but I still don't think I'd serve them if there weren't enough to go around.
Also is it ok to refill trays from the kitchen?
Does this mean that you'd take an empty tray back to the kitchen, fill it with more food and bring it back to the party? Sounds fine to me.