I know in ranked games, a characters elo determines the order of champion picks. In non-ranked draft pick, pre-arranged teams have priority. Does team size, elo, or anything else affect the order of champion pick?
AFAIK, the ranking only considers premade teams vs. solo queuers. Premades get higher priority (order randomized within the team), then any stragglers get randomly thrown in underneath them. This is why you end up in the bottom 1 to 2 slots so often if you solo queue for normal draft pick games - you're getting matched up with a pre-made (or 2).
I don't think that there's been any official word from Riot on this though, so this is all conjecture.
Pick order is random now in draft/rank mode. This was done, I think, so people can't easily recognize duo queues and get upset before the game (obviously riot doesn't say this officially).
The randomness does have some non-random aspects though. If you are duo/group queueing, the captain of the team will get swapped in for first pick(and banner) if any of the team is randomly put in that first spot and they will replace the captain in whatever spot he is in.