With a 4 sided die, the game will end on the first turn about 6% of the time. The Scholar's mate takes 4 moves. And if the other player rolls a 1, then it's a win for white.
Yeah, kings playing en-passant, rule 2, seems useless. The king moves in all eight directions, you can just like, take the piece. If the two kings try to pass each other, the passing king will check the opposing king, end the turn, then you just take the piece. Unless you're automatically checking the piece when your turn starts, in that case, it's just a stalemate as each king is checking the other at the start of the turn and ending the turn.
Third rule is just promotion but for all pieces. This is just saying "and everything becomes a queen".
Fourth rule is just a more relaxed version of castling.
Fifth rule is to just get new pieces on the board, because otherwise, the game ends in a draw. No one is going to willingly put their king next to the other king. Also "starting position" is kind of nebulous. There are 8 of them. Do I get to choose which column to start the pawn on? After that, it's just a race to promotion.
The rule about allowing another turn after check just extends the game. And unless you roll a 1, you should be able to get out of most checkmates.
This is adding more rules, not adding complexity. Every rule is to make the game easier, removing restrictions. The die lets you get around the 1 move per turn rule. Rule 2 is pretty much for decoration. Rule 3 gets you faster promotions. It's about who can get to the opposite row first. Rule 4 makes castling easier. Rule 5 is about not wanting a draw. Because with Rule 1, sometimes the king can take a queen with no assistance. This is a game the creator thinks is brilliant and thinks shows how smart they are because they are virtually undefeated because all of their opponents eventually get tired of playing because it's just so tedious.
It's like all the rules added to Monopoly. They're all to make the game "more fair" or "better", but all they wind up doing is making the game take longer because no one wins.