Before I read your post, I had never thought about this, so I tested it in my game. The same thing happened, the recipient of the gift didn't have it in his inventory when I switched to his family. So I'm pretty sure that's how it's meant to be. I don't think it's a bug.
Your second question, I have never seen any interactions for sending gifts when you click on the mailbox, so I don't think that's possible. The only options that I know of, are for sending love letters, and kleptomaniacs can return things they've stolen through the mailbox. That's only symbolic though, because I don't remember ever having seen any of the stolen items actually being returned to their rightful owners.