I kinda like your first idea and think maybe something could be done there. As much as 50% of each league's rewards get shared evenly between everyone who beats the first map. The remaining rewards are then payed out as currently is setup by total score. This I think would encourage more players to at least do the daily free shot.

I kinda disagree with your 2nd point though. I have 1 yes and it can make a huge difference yes, but it is not an I win button. I've stupidly placed it more than once and watched it get blitzed. I've also replayed some days 3-4 times and scored higher in later matches when the hero was on cooldown.