My advice for helping him calm down is to not put up the gun immediately and take some deep breaths. I tend to get excited when seeing any deer (doe or buck) so I just take it easy and breathe deeply and gain my composure.

On the second, it definitely sounds like your cousin hit the deer. If it fell where he shot it and also laid down 30 yards from where he shot, it's probably in the woods somewhere. He might have gut shot it and it hasn't started bleeding out yet but is hurt.