Comparing Joy with Happiness:

"Therefore, since we are surrounded by such a great cloud of witnesses, let us throw off everything that hinders and the sin that so easily entangles. And let us run with perseverance the race marked out for us, fixing our eyes on Jesus, the pioneer and perfecter of faith. For the joy set before him he endured the cross, scorning its shame, and sat down at the right hand of the throne of God." Hebrews 12:1,2

I think it is possible to have both joy and unhappiness at the same time. It was certainly not a happy thing for the Lord to be on the cross but he did it for the joy set before him. Also a believer could experience joy and unhappiness at the homegoing of a believing relative...joy that they are with the Lord but sadness in missing their loved one. I would say that happiness is an emotion that results from "good" external stimulus while joy is an attitude that is the result of inward choices (and the Holy Spirit's work) in spite of circumstances.