No; all your states allow it to be done in a ring if the ring has an extra lower bottom rope. That helps keep fighters from being pushed or forced out underneath, falling onto the floor or tables surrounding the ring and hurting or injuring themselves. A lot of your newer boxing rings are made with an extra ring rope 12 inches off the mat now for this reason. If you look at companies like Ringside you will see that they now manufacture and sell traditional boxing rings as well as boxing/MMA rings for this reason along with mixed martial arts rings and cages.

They are moving to a cage, but it’s not fully implemented yet. The reason is that the traditional rings with rope was easy to slip through, tangle up, and grip. The fights were constantly stopped in ground fighting situations to get away from the ropes. By using the cage there is less need for resets and you can actually use the cage walls tactically in offensive manner.(with ropes it was used tactically in defensive manner)

No, most of MMA competitions around the world, are done outside of a cage...It can be the same as an Olympic type of wrestling, it can be a ring, it can be anything.

Nope sometimes it is in a ring.