Getting Kayn missions done is a pain

Just give it a couple weeks and he'll turn into an average champion. The hype will die down, and judging from his current strength, he won't be very popular.

Biggest problem I have is the one where I have to win a game as Shadow Assassin lul

I haven't seen many dodges because of Kayn honestly (anecdotal evidence ofc, plus I'm from EUNE so not the same playerbase). IMO blind pick is the best way to complete the missions(not being banned aside), when I played Kayn I got Shadow Assassin always while enemy always went Rhaast (since it's considered better+you almost always get it first) and got the 5 games missions done fast, other just kind of came automatically through playing a few games.