Functionally, they're master and apprentice. But in truth, I think they have a deeper relationship than that. Zed sees himself in Kayn. An orphan with a gift and a drive that others can't control, no matter how much they try. No matter how much Zed tries, in Kayn's case, lol. I think Zed might also see something he could never be in Kayn. Kayn's link to the shadow is deep, for whatever reason. Maybe Zed hopes this is a sign that Kayn could do better than him? Ultimately succeed him? And he may have other, darker motives as well.
I think Kayn's respect for Zed is just as deep. Unfortunately, he's in the process of convincing himself that the only way he can prove himself to his father figure is by becoming something more than Zed wants him to be. Stronger than Zed. Strong enough to defeat him, if necessary.
They have a bond, but it will be tested. (This was from the ask us anything for Kayn https://www.reddit.com/r/leagueoflegends/comments/6n39xh/waddup_were_the_rioters_that_did_kayn_ask_us/ )
As for Zed caring about Rhaast taking over Kayn's body I don't really know I couldn't find any answers but if I were to guess Zed either admires the power Kayn has to hold Rhaast or that he is concerned for Kayn's well being and tries to keep a eye on his condition but not actually showing that he is.