Vayne/Kennen/Riven are good carry characters imo

Mid the the hands-down most influential position in the game. Jungler is a close second.

As Mid, you have fast access to all four buffs, both big objectives, ganks and assists everywhere, and the potential to limit your enemy's vision by downing their middle-of-the-map turret and forcing their entire jungle to require wards.

As the Jungler, you can help any lanes that aren't going well, steal enemy buffs, give your team buffs, take objectives, and overall help and assist allies who aren't doing so well (or help an already dominating person win their turret so they can start roaming).

I have personally started playing AD Carries since I'm so far above my level of play in terms of last hitting ability, and the midgame domination you can get over your lane opponent is crippling to the enemy team. As long as nobody's exceptionally fed, you can probably win most 1v1's (and teamfights if your team stays in front of you, tank or not). After you have a little bit of feed, AD scaling is exponential and you can single handedly take down turrets.

Some people also like playing Top. I enjoyed it for a while, but you have to counterpick your opponent and deny them like a boss while not missing much CS yourself in order to bring back a potentially bad midgame if your team is feeding. Top is also the most liable to get ganked, so you can waste the enemy jungler's time if you can keep ahead of them both.

I agree that mid is the most influential, with top and jungler being behind them. You already seem to have a solid set of AP mids (Ryze is another good one as he can build pretty beefy and still dish out damage), I'd recommend getting good at some top lane bruisers like Riven or Lee Sin. If you can dominate your lane, you become virtually unstoppable quickly.

Just be sure you're prepared to not rely on your teammates. This means really good positioning with your AP characters, and building tanky DPS on your tops, because if you're the only one doing well on your team and the enemy has half a brain, they'll CC and shut you down first.

There are a lot of theories on how to get a better ranking in solo queue, and for the most part they all work in different ways. But they all boil down to the same thing:

Find something you're good at and keep doing it well.

Some people get really good with a single champ and just play them over and over again, making sure to crush.

Other people get good at a particular role, and play that in order to win.

Context switching between every game and trying to fill the "last remaining role" in my experience isn't as important. If the last role left to fill is support, and all you know you're at your best when you play Ahri, by all means play Ahri. There's no reason to feel pressured into playing a role you're no good at.

Those are my 2 cents.

Oh, keep in mind, I'm not a pro at ranked either. I was doing the "constant context switching" and "Fill whatever role possible" and it didn't work out for me. Maybe others have better luck with it.

tl;dr find something that makes you look good and do it over and over again.