I feel like we should get one free dodge, without an LP loss.
I've had to lose LP recently due to one person (For example) being last pick, and letting it random for them.
"I did it on purpose, I don't want to play support" he said.

The AFK/random boot is a start.
However, I believe a vote/kick should be implemented (3 votes to kick, or duo's can abuse this by not having their friend being booted as the fourth), or something similar to the current game acceptance.

What I mean by that is, a confirmation of if you're there before the game starts.
Just like before we enter Champion Select, we must confirm we're there. If not, we're kicked out of the queue and forced to wait again.

This should happen before the game starts.
It would prevent those AFKers and d/c's to start. I can't tell you how many times someone's just disconnected in champion select, and then never showed up to the game (Or fifteen minutes in) and lost it for us.

People may say they use this time (After picking their champion) to go to the bathroom, get a snack, etc. But I think it's a small price to pay to keep trolls, AFKs, and d/c's out of the games.

I think the dungeon finder would be nice, but in turn would result in longer queue times (which people complain about already at times) and would enforce some kind of meta, which I didn't think Riot wanted.

I almost wonder it we could combine these things, not even just do one, although I'm sure they'd want to add only one at a time in order to test how things work.

Of all 3 of those methods listed, I think that a vote-kick is the most needed right now. I understand that people don't want vote kick to be used to kick players going against the meta or something like that, but the fact is that if it's UNANIMOUS that the other 4 players on the team don't want to be subjected to playing with that person, they shouldn't have to spend 20 minutes in a game that's not enjoyable or face a dodge penalty.

The fact is this: If there's a dodge penalty, there needs to be a vote kick option. Otherwise you're forcing players to suffer consequences for trolls and non-team-players. They either have to face the dodge penalty, or take up a "don't let the trolls win" mentality and play out the game anyway.

I think role queuing will always be a problem for a game like this, because it's very hard to avoid supporting one team comp over another. Queueing needs to be as vague as possible to allow for unique team compositions.

Although there are some decent ideas on that subject in my thread on the topic here.

Solution 2 is horrible, I don't want to see it happen because it enforces the current meta which pushes out uncommon strategies like running tanks in botlane, not to mention most low level players can't jungle (players become more powerful as they gain masteries and runes, but the jungle is just as hard on level 1 players as it is in challanger tier)

Solution 3 I don't see working at all. It'll just make a ghetto of bad players who don't have any incentive to improve, and their attitudes will probably rub off on other players (especially if they're using the same LoL-related news and fan sites). Dota2 has this and their community is a lot worse than LoL's, which is no small feat.

Solution 1 looks like the best of the bunch, I don't have any major problem with it assuming it's implemented correctly. If it's implemented badly it can be horrible, but that's what PBE is for i guess.

if you pick #3 you are matching people who have a history of raging (be it once or more) with like minded people. what you would end up with is a downward spiral of hate and rage that would end up getting those players most likely banned. while this might seem a good thing to most people, surely it would serve the comunity better to try correct these people or control them (with options 1 and 2) than just giving up on them. equally if someone had a bad day and ended up in what we can call the 'toxic pool' it would be supprisingly hard to get out of. additionally being stuck in a game for 30 mins or more with 9 people who are very toxic would subtract a lot of the fun that can be had in this game

Edit:
a further idea would be based off of the honour system, there is a large potential for abuse and negative environments for these options but you could have a team captain based on positive play, (leadership honour i guess) who could be in control of the vote kicking.
oh and adding pregame chat to tribunal reports. and some sort of notification if one has been reported so they can be a little more aware of their actions.