I am sorry for the lack of video as I don't know anything about technology, but I do know my fair share about this game. When you enter the sort-of fighting pit in the arena, there is a pretty narrow walkway followed by some stairs which lead to an opening to the gate that is closed until the announcer is done speaking. However, if you turn around after going up the narrow stairs, you will see the doorway which you went through which is oddly shaped. If you go to the left side of this doorway facing the exit to the fighting pit and you jump up the doorway a bit (this requires a bit of acrobatics but nothing too bad, I will get into that later) you will clip out of bounds and from there you can down-warp into the middle of the arena. This alone saves time waiting for the gates to fall, BUT you can also maneuver yourself up the side wall near the opponents gates and get over it to kill them from there. A great way to get your acrobatics up quickly to be able to perform the clip properly is to trigger the announcer to start speaking and as soon as that is done, go back down the stairs and get your character stuck against the wall-ceiling near the first stair. The best way to get the wall clip consistently is to jump facing the doorway then turn 90 degrees to your right so you're then facing the stairs that were to your right. Holding forward and left from there gets you pretty consistently out. I have a feeling this will save at least some time.

That clip is known about, I did it in my run for most of the opponents that take more than one fireball to kill. For the ones that do take only one hit it's actually faster to do a save clip directly through the first gate, then try to shoot a fireball around the edge of their gate, then clip back through. Acrobatics isn't too much of a problem since before going to the arena you get skooma which makes you fast, which makes most clips really easy, like for the clip you described you wouldn't have to jump for, when you're on an 8 stack of skooma you can just keep ramming into that wall until you go through.