Brainstormed a few ideas, and came up with some that might work or could be expanded on:

1. A pre-game lobby, possibly with vote-to-kick implemented. This gives more time to sort out picks and bans.

2. Queue people in together, and then look for 5 opponents. This may cause problems but it would inspire players to improve themselves in their roles.

3. Preferred role icons with everyones summoner names.

4. Matchmaking based on what role you play more often using your recent 10 - 20 RANKED games.

5. Bar ignored players from playing with you. They could possibly still queue against you.

6. If a player fails to connect or disconnects within the first few minutes of the match, allow the game to be disbanded with the elo loss given to the disconnected player.

Feel free to expand on these ideas and give feedback.

Just for clarity (I don't want to assume) could you list what problems these solutions are aimed at?

Sure thing.

1. A pre-game lobby, possibly with vote-to-kick implemented. This gives more time to sort out picks and bans.
I image that a pregame lobby would give a team more time to figure things out, allowing for early game strategy to be discussed. If an issue arises or someone becomes problematic, and the team decides that they do not want to play with this player (be it they are showing signs of toxicity or trolling) they can choose to disband that player before they even get into champ select.
I honestly believe that if 4 people do not want to spend 20 minutes to an hour with a certain player, they should not have to. Games would in turn become more pleasant and structured.

2. Queue people in together, and then look for 5 opponents. This may cause problems but it would inspire players to improve themselves in their roles.
Similar to the first idea, this would queue 5 separate players in together for a ranked match and allow things to be sorted out before finding a team to play against. Players could dodge from here without having to worry about a penalty.

3. A preferred role icon would be more of a personal thing for each summoner, and allow players to discuss what they plan on picking with a better starting ground. There can be a bit of chaos in the time it takes between bans and picks, and this would help to calm that chaos down a tad.

4. Matchmaking based on what role you play more often using your recent 10 - 20 RANKED games.
I actually think this is a solid idea. Say you play a lot of ranked games as a jungler; you would end up getting queued with people who instead frequent mid, top, and bot without having to worry about fighting over roles. You would also get matched with players who have more experience in that lane, instead of just being forced to play it because they had to.

5. Bar ignored players from playing with you. They could possibly still queue against you.
I have lost due to a really toxic player, and I'll requeue again, only to get them back on my team. So i dodge and have to wait 30 minutes.

6. If a player fails to connect or disconnects within the first few minutes of the match, allow the game to be disbanded with the elo loss given to the disconnected player.
Kinda self explanatory, and is implemented in other games. Idea should be tweaked if used.

a few of your ideas have problems that, I think, would prevent them from ever being implemented.

I LOVE the vote kick option, its been going around and I HOPE riot will set it up soon enough, with some kind of system to prevent it from being abused. The pre-game lobby idea, however, gives whatever group of five gets together first an advantage. It gives them more time to discuss and organize things. This, like I said, would be an unfair advantage. Same goes for getting a team of five together before you find anyone else to play against.

number 4 will also likely lead to longer queue times. People who play ranked (and are not professional players) should be prepared, willing, and able to play multiple roles and deal with it if their preferred role is not available.

just my thoughts.

Furthermore, You'd need a way to keep 6. from being abused. i.e.: me and my buddy are playing ranked game. Team starts doing ****ty I alt f4 my game, take the elo loss so my buddy doesn't have to , and he does it the next time it happens.

people don't often get banned from league for internet problems, if you catch my drift. (and no, I don't do this. never have and never will... just saying it would be abused to all hell)

I do like a number of ideas, and I'd like to add to them, if I may.

If you have ever played World of Warcraft, you would be familiar with the Dungeon Finder. This groups 5 random players into a team for some PvE action. When you enter the queue, you select one of three roles: Tank, Healer or DPS. A team of 5 consists of 3 DPS, 1 tank and 1 healer. A team is not complete until all of those roles are filled, and once they are, they begin the Dungeon.

I think a similar system could be used that reflects many of your ideas. When queuing up for a match, you can choose the role you'd wish to play - Top, Jungle, AP mid, AD carry or Support. This directly relates to ideas 3 and 4:

Once the team is made, they can enter a "pregame" lobby as you describe, which operates similar to the Custom Game lobby. That is, you can chat with your teammates prior to clicking "Start Game". Of course, for matchmaking, there is no "lobby leader" so it would have to be modified to have the "Vote to kick" function, and also a "Vote to Start" or a "Ready" checkbox so that once all players are "Ready" they go into the queue and begin looking for another team to play against. This directly relates to your ideas 1 and 2.

Finally, idea 5 should already be implemented. If a player is on your ignore list, I don't see any reason why matchmaking would put this player on your team. No additional ideas from me, just make sure that you never see an Ignored player ever again

I think idea 6 is a decent idea, but may be open to exploitation. To add to the idea, what if there is a "Vote to Draw" function in which both teams can vote to end the game within the first 5 minutes if there is a player that cannot connect to the game, or anything like you described.

That's the best I can do to enhance your already great ideas.

Like I said, the final idea may be open to exploitation and may cause more issues than it solves, but the other 5 ideas are great.