Thank you for a response.
ad 2) I know it is not currently possible, but I believe implementation should be easy
ad 1) I do not think that it would be very difficult. In endgame and just before endgame the potential to hunt the king on empty board definitely exists. As game progresses, it will be easier and easier, until one player does it. I don't think that setting up the attack were one piece sacrifices itself for a check, followed up with two check by heavy piece, especially queen, would be difficult and rare. I could make up hypothetical scenarios for this. And sure, you can protect yourself form this, but that would require using pieces defensively to protect your king. And that means they can not be fighting in center, and you may then lose to pawn push on the other side. If you chose to be too defensive, using pieces to protect king and leaving rest of the boar for opponent, opponent does not even need to protect their king because you are in no position to attack with all your pieces back.
3+ in row variant

Simple,
like three check, but three checks must be made in row (in three subsequent moves). Whenever you do move that is not a check, counter on the opponent king is reset to three
I think that in comparison to ordinary three check it would be more strategic, attempt to hunt the opponent's king would require more planing for the attack, but it would still require elevated kings safety and basically eliminate draws form the game.