"2)Make promotion cycled! Kings do turn back to pawns. However, kings still
can't move into check (but thereafter became none-royal again)."
In theory the king should be allowed to move at check with this rule (but can't stay into check).
The idea of check is to make impossible to some player make some mistake move that will end the game.
If the king will turn into a non royal piece, then by making the moving and puting this non royal piece into check, he will not end the game and lose by a mistake.