Yes, you can buy him a gift or something, that will make him happy definitely. However it's not like you have to do it, there is no compulsion or a rule as such. By choice if you want to buy something, or if you believe that's right then you can go ahead and do that.
Sure, but make it a small gift. Don't make it seem like you're trying way too hard or make him feel bad if he doesn't give you a gift. Possibly ask him if he wants to do presents? That'll make it so there's no confusion.
while this sounds really bad usually ( as a girl) ill cook my boyfriend something and as we are eating dinner he will give me his gift ( usually a teddy bear or jewelry) and after dinner ill give him mine.. which usually happens in the bedroom. He seems to like it and I don't mind