(!) equipment restriction challenges- Guessing that's go into an area with or without a certain item.
Nope, means going throughout the whole game without changing equipment, or only using/not-using certain equipment. Could also be not using any of the equipment you're given through story. Could also be using any equipment but without customization/alchemy. Things like that.
(!) low level runs- Is trying to survive a high area with a low level? Or just to see how fast you can get through one?
Finishing the game at the lowest level possible. Try to beat my record of Rebirth at lv38, Reminisce at lv71, and Redemption at lv109 ;D
...Or beat Master ZED's records. They're insaner than mine X_X Anyways, it means that...not an area in specific, but the whole game, as low as you can.
(!) skill trigger challenges- No idea
Not using the Skill Trigger in any battle. That also means no Rengeki. The Arena is never easy anymore if you try to take on this challenge, unless you're several levels higher. But overlevelling is never fun.
(!) We've even got creative in avatar battles- This one sound interesting, any chance you can go into details?
Finishing Avatar Battles without receiving any damage, or not using shots, or ONLY using shots, or the fastest possible, or more than one of those "challenges" at once. You can actually get to learn alot more about how the Avatars fight and their behaviour while doing challenges like this.
Not sure about what he means by Team Attacks '8D lol
And here's the link of a few YT channels with videos showing several of these challenges I WARN YOU THOUGH, these show battles against bosses from late in the game, final bosses included. So if you don't wish to be spoiled, it would be best to stay away from them until you're done with GU the first time:
My own channel, with Low Level boss battles, some Avatar battles, and a few other curiosities.
Master ZED's channel, which more more boss battles, all Avatar battles, and many other things.