Strange, it seems like Yato really is loved by his father. I thought the relation was st...

I don't feel like he is actually loved by his father. I just think he wants to protect him because he is the result of his strong wish so he probably needs yato so that his wish can be fulfilled...
But even if he does care about him, his love sure is twisted...

Didn't heaven kill someone precious to him? don't really remember but heaven probably wronged him somehow

Well, it's a strict form of love. A weird one, not pure and unconditional. I don't think he ever thought himself that he would go this far to protect him.

Remember, how he said that Yato can't reincarnate, that it would be the end of him. That's his motivation, at this moment, not his desire of use for revenge, but of saving him.

I do think Heaven killed someone close to him but I don't know who. We weren't shown much of that flashback. Was the person human? A sibling? It certainly seemed to be a male someone, right? With black hair?

I'm also curious how Yato came to be and if it is tied to his father's relationship with the one who died.

I am certain that person was a human. Remember the time in Yomi? Nora said how he got out, somebody called his name, so it must have been a human. I think He got that brush before his beloved has/have died, before he started to hate the heavens, before Yaboku has born. And to think that his weapon is monk's staff, it should mean that he was kind of maze, sorcerer. If the Nora know her true name and still going on than she do had some connection with Him before she died as a child. Should be that He lost his beloved woman and his child ad Nora is this child and the gonna get revenge together?
For Yato to eliminate this person he will need Yukine to get t through his previous life and still remain shinki, I think. But I hope that won't be necessary.