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Right now there are three stages to the champion select, Ban, Pick and Free. My suggestion is to add a fourth one, before pick order is established, that allows people to sort out a) who's going where, b) what should be banned, and c) if there are any trolls so that dodges can be easily made. I propose this part last five minutes or until everyone confirms that they're ready to proceed. I also propose that this include a way to look at a person's match history, so that when fights break out over, say, who should be top, they can look at the other person's stats and see that, yes, guy A is better at that position than guy B.
I think part of the hassle is largely because there's really no time to settle all of those things easily. You'd have to have five conversations at once; what do we ban, what's your position, can first pick get me Morgana, etc. It wouldn't solve the troll problem but I think it would ease the pressure that comes from having to choose roles so quickly, plus it would end the mindless debate over calling vs. first pick.