Many successful pieces of furniture have joints which are like you describe. For instance, look at any wood door. The grain direction for the rails is horizontal, and the grain direction for the stiles is vertical. You can get away with it if the joint isn't too long. I wouldn't be too worried about a 7"-long joint. If you want to be really paranoid, you could put glue on only the middle five inches or so.

Yes, a mortise and tenon joint is a good choice here,