@tpr in other sport it's being used exactly so that initiative is swung back and forth between team A and team B or player A and player B (tennis for example and more recently soccer).
In chess it is well known that white has an edge over black (55 vs 45 win rate).
Yes, but I think this variant gives the advantage to black. for example 1 e4 1...d5, dxe4 threatens 2...Bg4, Bxd1. White is on the defensive right away. White has 2 moves to defend, but then come 2 black attacking moves.
It is a totally different game, where Bishops and Queens get much stronger. The main point is that one piece can make two moves at once.
Also, forks are useless, as you can make two moves. So are skewers and pins, as you can move the pinned piece and the piece behind it. Also, I have a question. If your opponent does a double check, can you capture both pieces on both your turns, or do you have to move our king?