I don't think it's just a guy thing to want to actively want to pay. With my ex I would always genuinely prefer to be the one to get the bill, not because I wanted to "wear the trousers" but just if I really like someone, money is no object. We compromised by taking it in turns, but sometimes there'd be times where one of us really wanted to pay and treat the other person, even if it wasn't our turn (not that we kept track). I think it's generally quite a polite and generous thing to do and if a guy wants to pay he shouldn't be made to feel bad about it, let him enjoy the moment. It feels good to treat someone you like. But no, guys don't have to pay for the first date but I think a lot of women appreciate the offer because it kind of indicates that you see a future and that you don't mind paying initially because it'll balance out when you go on more dates.
When you're as young as 16, you don't tend to have a lot of money so I think it's less expected to regularly pay. But I think on a first date, if you've specifically asked them to go somewhere with you, the polite thing to do is to pay their half, or at least each buy your own cinema ticket but you pay for the popcorn. It's those little gestures that can show you're serious about her and like her. They're not necessary but they kind of make it feel more date-like.