Illaoi x Karma

1. Yuri. While not bad, it's a terrible pairing sequence for a fanfiction.
2. Their relationship would be based on willpower. If you wanted it to be successful, they'd have to go do something like learn the secrets of the Dying Will Flames (from Katekyo Hitman: Reborn!). That wouldn't be a bad story at all (for a crossover), but if you wanted it to remain solely League of Legends you're not going to have a very interesting story. 3/4) It's League of Legends, and they're going to be doing a lot more fighting than making out. Even if they were interested in each other, the nature of League of Legends wouldn't really let them "explore" if you wanted it to be interesting...unless you went Adult-rated, but if you wanted it to be rated for younger than 18 years it's not going to be interesting other than fighting in the League. Which has been done. A lot.
3. Illaoi wouldn't be satisfied with just Karma, if at all. She says it herself, to be satisfied is to "not be in Motion".

As a pairing itself, I could find a lot of better pairings. For Karma, Twisted Fate or Garen would be good, or Riven or Ashe for the female route; for Illaoi, Gangplank or Braum, or Katarina or Sejuani. I don't see Illaoi and Karma, though.

Meh, still shipping X

I feel there is a connection. Not a physical one.