For Shen, I imagine Kayn knows him to be a former friend of Zed's, and thus recognizes him as a peer, even if from another faction. When he kills Shen in-game, he says "consider this a debt paid to a friend," so it's clear he acknowledges the past between Shen and Zed, and knows that they were once close.
Kennen and Akali are both slated for reworks, so it's too soon to say anything about them.
As of right now it's currently unclear if Zed still has any standing with Syndra since his lore got rebooted, so I'd say Kayn is probably neutral to her.
As for Zed... Kayn DOES admire him, but at the same time, he wants to get even stronger; strong enough to surpass his master and become the new leader of the Order of Shadow. We currently don't know how the order of succession works in this organization (especially since Zed is their first and only leader so far), so it might be as simple as naming Kayn his successor, or it may be a "you must defeat your master if you wish to be the master" situation.
Lastly as for Rhaast... He and Kayn are enemies fighting for control of the body, but I don't think they hate one-another. They just recognize that their struggle can only have one outcome, and one of them will have to die in the end.
