1. and 2. A night out and a day out are the same; they just take place at different times. You spend the day/evening out of the house, or out of the office, doing something fun.
3. A cookout is when you cook outside. You may have a grill or an open fire. It implies that there are several people there.
You're hanging out, for sure. It could be a night out. If it's a break from your usual routine, and you planned it in advance, it could be a night out with the guys.
A night out (or a day out) is a little more special than just hanging out - it's a varation from what's usual, it's usually planned, etc.
On the other hand, if you stop by the bar after work and see your friend, whom