I have never wanted to play ranked, because I knew the horror stories of "elo hell" and then the whole "it doesn't exist" replies, and all those typical scare stories of bad players. I have been told that there are losing streaks and winning streaks when you play ranked.
Personally, I believe in ELO hell, as of now.

I will try not to complain about my teammates, because everyone knows about those 0/19 urgots and those 0/11 katarinas, those 1/11 vaynes, ect. I will, though, admit this is what I am asking for help countering.

How can I carry so hard, that my team's inevitable failure will not cost us the game?
This is what I have been doing/what I know so far:

1: Always pick a strong character that can carry if needed. Pick someone that can snowball. My choice was Jax, because he can top-lane(no teammates needed), carry, and has a lot of burst so can get fed easily. He also has tankiness so he can last in many engagements.

2: Gank mid whenever possible as top, if top. Since I play top almost exclusively now, as it gets good money and has a character type that easily snowballs, I gank mid often. I feel like this does not actually effect the game too much. You can put out the campfire but the forest fire still rages. Something similar to that.

3: Ward everywhere. I know this one from normal que as well. Just buy wards, put them everywhere. Its the most important item yada yada.

Of course I don't want tips for the game in general. I want tips that can help me specifically at this level of play. I do well every game, but that isn't enough, so clearly there are some tips I am missing.

TLDR: Give me every tip to carry a ranked game, outside of picking a strong snowball champion, playing well in your lane, warding, and ganking mid often.

huge tip from my experience, play Draven, bot great adc late game. don't initiate more than 1v1's and you will barely die. gl man

I don't know if this is a smurf account or not, and I appreciate the sentiments, but that advice never comes into play outside of laning phase.
Even if you get incredibly fed, the game comes down to teamfights, pre-baron/dragon-dances, and tower fights.
1v1 is rarely an issue, especially as Jax. Jax is one of the 1v1 kings.

Well for starters, top isn't where you wanna be. You'll have the least impact throughout early game, which is the most vital part in solo queue. You want to be going mid or jungle, as those two positions have the greatest potential for affecting the outcome of a game. Pick a mid laner that can gank like crazy and snowball off of it, feed your laners, and proceed to win. Ideally, get the other team to cry about how well they were doing, but their "noob mid laner fed and didn't call MIAs". Just get really good at one or two mid laners, and proceed to carry. Or play jungle. That's basically all I've done so far, and I have a fairly solid record.

What characters do you think would be best for this? I was thinking for jungle: shen, lee sin, skarner, jax, or for mid lane: kassadin, ahri, gragas, katarina.
Side note, our elo is very similar.

i would say no...do not buy wards..information is only useful to people who can make good use of it. If you are playing with bad players...then the wards will not help your team. Think about the team. Yes you will have very bad players who do foolish stuff...and in fact they will ragequit sometimes when you dotn cooperate with their bad decisions...but if you jsut continue playing solid, you still can expect more wins than losses. Dont be afraid to wander mroe than you normally would to help out bad players in other lanes.
mid would be a better choice than top...bottom lane is important

jungle shen is great for helping to carry a weak team...you can prevent feeding, you can split push, and you can cause serious pain on the enemy team if you get good income.

My ELO ain't great but it is climbing. Take this advice as you will.

To me it feels like the key to carrying is focusing on taking down towers. You want to get a few kills on your lane opponent and then bring down the tower ASAP. Then move into another lane and bring that tower down.

The early goal is to secure a money advantage for your team. Even if you team is stacked with noobs if you knock over 3 towers before the enemy gets any they may do ok just because of the extra gold you get them. An earlier BFS can make all the difference for your AD carry.

Less towers also makes the enemy champions more vulnerable. Ward offensively and pick off any champ that wanders too far from his friends.

Mid to late game you need to decide whether your team is capable of winning a 5v5. If yes then try and get your whole team grouped and just push mid for victory.

If no then you need to split push. Drop wards so you'll see anyone coming as you try and take down top or bottom tower. If you see only 1 person coming you kill them. If you're fed like you should be you can probably kill 2 if they come as well. More than that you probably need to run, but the rest of your team is now at 4v2 or better. Get them to push the opposite tower. The enemies in your lane will probably run back and you can go back to killing your tower.

Yeah... I don't play ranked much. I have just as much as fun in normals, and never run into these "horror story" team mates. I have no reason to play ranked. The stuff I say is just based off of normal drafts, what I've heard other players say, and my little ranked experience.

For jungle, take someone who can gank. A lot. But still have at least some carrying capacity. Skarner is an amazing choice. He's currently up there as my favorite junglers. He takes the concern of "Oh! How to kill the ADC?" right out of his team's hands, and instead drops their ADC into them. Shen would also be good if you can sneak him though the bans. Just make sure you're split pushing hard. Jax needs too much farm to be effective, and Lee falls off too hard, so I wouldn't recommend them. Unless you're a diamondprox-level Lee Sin.

Mid. Hmm. I avoid mid like its the plague. I'm a crappy mid laner. But I can play LeBlanc very well. She's incredible. You have no idea how much morale is destroyed by a fed LeBlanc killing your ADC again. And again. And again. Seriously. She just makes the other team feel bad. Kass is also good, though he's weak early, which is something I tend to avoid in mid. I've been seeing a lot of Kats carry recently. I'd say she's a solid bet. Ahri and Gragas are always nice, and will almost never be a bad pick. Karthus and Twisted Fate are also good, because "global" presence is OP.

Top lane can only carry if the rest of your team does somewhat decent. It's very isolated so there's only so much you can do.

The best way, contrary to what a lot of people think, is to help your teammates as much as possible and work together. Now, realize, if you see someone is bad, avoid them. But almost ALWAYS there's at least 1 other decent person on your team, and if you make it a point to work with him, it increases your chances of winning a lot.

To this end, you need to pick either a jungler or ap mid. These are the people most free to roam and help the GOOD players and keep the bad ones from feeding too much. I will try to save the bad ones from feeding but I won't risk my life for them.

Once you identify the good players, help them in ganks, secure objectives with them, even pick your spots and invade the enemy jungle with them. 2 people can often make huge plays and easily steal buffs and bully the enemy jungler whereas a solo can only do so much (notice how this is all easier if you play mid or jungle?).

One way to almost guarantee kills? If you see either mid or jungle going to gank, go with them (this applies mainly when you're playing mid or jungle, which is why I say to play those). Not only can you help secure a kill, you'll also rack up assists and gain the trust of your team, and they will be more willing to work together and stay positive as a result.

My last tip would be, get good at knowing when to push lanes. If you ward and have good map awareness, its easy to keep your lane pressured and the creeps near enemy tower, making your opponent lose cs to tower and forcing the enemy jungler to waste time trying to gank you. If he's ganking you, he's not ganking your (bad) teammates and losing jungle farm. I often use this strategy when playing as morgana, since she pushes very well and can easily escape ganks. This only works if you have good reactions and know how to escape ganks from average players, however.

Find a champion that has a gamebreaking carry status, dont blame, learn a wide variety of champions to increase your knowledge and understanding of the game. Elo will come with a higher understanding of when to do dragon/when to baron/take objectives.