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Yeah, there are a lot of ways that the game could go about handling that situation. I think the only reason this problem occurs is because it doesn't appear that the game recognizes when both players die at the same moment; instead it looks like the game checks player 2's deaths first, then player 1. This would explain why both players  seemingly respawn at the same moment but with consistent differences in how they respawn. As for gameplay, It might just be best not to give either player a sword ever in this situation. Although the situation is a rare occurrence, when it does happen it can be the difference between winning and losing a room. I think that if the player with the least deaths were given priority of the sword, it could create a snowball effect to give that player the advantage (ex. a player has several less deaths than the other and is rewarded with a sword, which gives them the advantage to further the gap of their death counts). When playing it is more fair to put both players on an even playing field, giving them equal opportunity in the fight and keeping the game short term.

As for the infinite height, yes it can be achieved!!! My friends and I discovered double-jumps and love them as they add another level of depth into fights, but sometimes it can get a little crazy. We found 2 main ways they can be done. It is completely impractical but pretty cool. Here is a video demonstrating: