I don't know what causes the enemies to vanish, but it happens when the player jumps at correct time/location, when an enemy enters the playground from a door. The enemy probably gets confused due to being unable to target the player and tries to perform some special move that teleports him off-screen. The HP of the vanished enemy does not change, so he is not damaged in any way. Oh, maybe it is related to the fact that if a boss is killed and there are still basic enemies around, the basic enemies will flee and then vanish. Maybe the untimely vanishing enemies erroneously get to the "Oh no, they killed my boss!" state when the player is in the air? Just guessing.
Well, I tested doing the first fight with 2 players. There's not so much lag and the 2nd player starts ahead of player 1, saving time immediately. Player 2 causes heavy lag in the beginning of level 3, so he should probably suicide in the end of level 2. Or if the player 2 starts ahead of player 1 in level 3, then player 1 should suicide instead.
The first fight with 4 enemies takes less than 10 seconds, but it took me more than 1000 rerecords to do and it's not even close to perfect. The enemies had 19 HP in the RAM, but it was actually 20, because they die when they have -1 HP. Consequently, I wasted time due to bad planning of using different attacks. This is also the first time I did something with 2 players, so I'm not used to it. I don't know if I would enjoy doing this game for real, but I'll be helping as much as I can, if somebody wants to try it out. Judging from the first semi-sloppy fight I made, it will be an entertaining and fast-paced movie:
http://dehacked.2y.net/microstorage.php/info/1356681397/Double%20Dragon%20-%202%20Player%20Test.gmv