Is it allowed to bring cousins or friend which are minors with permission of their parents to camp in a tent?

I don't know that it would be an issue depending on the type of pass you have to enter the park. This is the site to Zion's campgrounds. You can probably find specific details once you select which site you will visit. Have fun!! https://www.nps.gov/zion/planyourvisit/campgrounds-in-zion.htm

Good !

There is a check in area before you can enter the park. You must register yourself/guests at that point

Yes. A great experience for kids & adults.