If a boy has asked a girl out on a date, he should pay. This is just the polite thing to do. If for some reason the girl has done the asking, then the responsibility of paying is hers. Under no circumstances should anyone on a first date reach for the check and demand the other party pay up! In longer-term relationships, you may find yourselves switching off occasionally, as both parties will be making plans, and this is fine. The EGs just think it's nicer to alternate than to have both of you reaching for your wallets every single time you go out.
When you are taking seats at a theater, concert hall, movie theater, church, lecture hall, etc., the girl precedes the boy into the row of seats. In a restaurant, when the maitre d' is leading you to your table, the girl precedes the boy.
When a girl is taking a seat at a table, the boy pulls out her chair for her. This is a maneuver that takes a bit of practice to be done gracefully, however. (The EGs have heard embarrassing stories of teenage boys who know they should do this at, par exemple, the prom, but get a bit over-zealous and yank the chair completely out from underneath their poor dates. Not a pretty picture, dear reader.)