If you want to make a contest, it's really up to you to sort out all of those things.
However, BricksInMotion has a bunch of official contest every year:
BRAWL (Brickfilm Rapidly All Week Long) normally happens in the summer holidays.
THAC (Twenty-four Hour Animation Contest) normally happens around the Christmas holidays.
Often there's an additional summer contest, like ToY (Tales of Yore).
If I remember right, there's also an Easter contest.
Now other contests, like Celebration, do pop up from time-to-time, and those are official BiM contests. Conversely, the 2015 Contest (interest in which seems to have all but peetered out) was entirely 'fan-produced' - the admins of the site had no involvement. I started and will judge the contest with a couple of other members, and other members will also be providing members, but it's up to use to do all of those things because it's not an official BiM contest.
Organising a contest is not as straightforward as you might think, though I appreciate that you have begun to give the processes of making a contest some thought. To answer your questions in order:
1) Up to you. You can PM them, make a thread asking for judges, YouTube message people...
2) I'm afraid I highly doubt Brickforge and the like would want to sponsor a small, unofficial contest, so it would be up to you to buy and distribute the prizes.
3) I'm not sure what you mean by this. Mods and admins normally discuss things in detail before they are released publicly, so I guess you could send your idea to one of us and we could 'review' it.
4) That's up to you.
If you think you have a really original idea for a contest, that's great but you might want to check with some older members first, in case it was done before you arrived on the site. I'd also, however, advise you not to make a contest, which may sound hypocritical having started 2015, but I'd note that despite having prizes and everything 2015 seems to have all but failed. Nobody seems to be making entries, and the thread has been dead for months. Other new members often try to start contests on the site, and this normally ends in controversy I'm sad to say (I will not mention names).
There's actually nothing to stop you making your contest right now, as a thread in General Film Discussion, but BiM will not sponsor it, and it's up to you and other members who might be interested to organize and purchase everything.
Sorry if this seems harsh, but this is the policy with contests: BiM has its official ones, and the rest are up to its members.
Now is also probably a bad time to start a contest, what with it being 'back-to-school' time.
I hope all of this helps. Don't be disheartened by any of what I've said.
"Nothing goes down 'less I'm involved. No nuggets. No onion rings. No nothin'. A cheeseburger gets sold in the park, I want in! You got fat while we starved on the streets...now it's my turn!" -Harley Morenstein