I've attempted my promos at least 5 times now, and every time I get at least 2 trolls on my team. And I can't dodge because that would count as a lose. What can I do to actually get a team that takes this game seriously in Solo Q?
Exactly what the first post said, you can't avoid them, but you can prevent those types of players from getting in your games by maintaining a positive attitude and if you can manage not reporting anyone after a game. I've found that when I report, when i rage, and when I generally just be bad myself, Riot places those exact same type of players with me. I'm not a fan of that idea and it leads to just more raging and the like. I try my best to be a good summoner, but damn Riot... this matchmaking is hell.
You dont, you're best hope is to dodge or hope you and the teammates that are not trolling can carry
The best strategy to avoid trolls is to watch for the tell-tale signs in champion select and dodge accordingly, or type "/mute all" In-game. It really is worth it to mute everyone, including your own team. Most important plays are communicated through pings anyway... From my experience, people rarely get on with the intention to troll, but people make them angry and they end up rage-trolling.
I suggest just muting everyone at the beginning of the game and not unmuting for any reason. Just don't get involved and do the absolute best you're capable.
You can't.
But you can start praying to Laguthulhu in the hopes that the servers don't fuck your promos over. 
There isn't a way to avoid them. The best "solution" is more to try and play the best you would and not get in the trolls way. Eventually they'll get punished and possibly banned, so you won't have to see them again. Also the odds of you playing with them again are incredibly low because of how many people play this game. The only time you will run into the same people repeatedly should only be at the lowest and highest levels, of which you are in neither, so it shouldn't be a problem. Just ignore them and communicate what you have to, pings being the best. Just don't overping, once or twice is fine. Play your best and just hope, that's about all you can do.
