Why would they want to execute a hero who defeated the Demon Lord?
This takes time to be revealed (3rd Arc), by basically, the world he was in was something like "Hardest Level", where basically the summoned hero starts with little to no help, and pretty much have an antagonistic way of treating the Hero, you could say it's a world where the Royalty WILL betray the Hero after he remains as the sole power against the Kingdom.
In the second arc, another summoned hero appears, but that one was summoned to an "Easy level" where not only everyone is friendly, when she returned to Earth, she was back to the same instant she was summoned from.
And yes, this Easy Hero is a she, and she is the one to explain everything (As Keisuke got no such info, in fact, I myself wonder how she got such info...
shrugs), and no, she is not part of the harem, in fact, she is the enemy of the arc, and after having been killed numerous times without even making such a fight against Keisuke, she gave up, and understood that picking up a fight against someone who survived the hardest level world was the dumbest thing she could have done (She didn't knew that before, although she knew of the level difference). She absolutely fears him.
Why did his crush have to sacrifice himself?
It's unknown as of yet in the novel (I read the raws and am way beyond the translation of the novel), however what we know is that, to summon the Demon Lord, the hero has to sacrifice that which is the dearest to him, and Elza did it. Supposedly, the status system that he still has access has her voice now.
I think the DK for the easy world was also explained, but I can't remember the details, I remember that she had to kill an illusion to reach the DK though.
WTF is up with his paternity skill that was mentioned once then apparently wasn't working on the little bitch that pissed herself after accusing him?
Only those who regard him as some sort of father can included in the skill, it's not like every daughter in the world would suddenly treat him as their father. Each of the heroines has a different approach to their fathercon, Angelica is to pamper her father, Ayako is obsessive, Rio is somewhat of a masochist, and so on. Also, the reason he got the Paternity skill is revealed much later, but Elza was pregnant, after over a decade trying to have a child, she got pregnant, and is also when the whole deal happened. The manga may not show it too much, but Keisuke was a complete mess, and he was even unshaved, as Angelica mentions when she speaks her mind at the end of this arc from the despair she had just faced.
In the other world, Keisuke was initially regards as a useless Hero who couldn't even summon a Holy Sword, he was thrown into battle and almost died, being able to awaken that sword in the middle of the fight, from there on, the King sent him to various battlefields, and from everywhere he was treated as Hero, even to his party members. Elza was a human that lived with Goblins, uncivilized, who Keisuke saved (Keisuke is also a Goblin Slayer, having exterminated all of them, for Elza). While other people saw Elza as a filthy uneducated human raised by Goblins, Densuke treated her as Elza, and Elza treated him as Keisuke. Elza is one of the reason he could still remain human.