and do not understand why I get matched with people who are not even ranked yet?
You say that like these people are worse than you. They're not. Unranked people can be placed anywhere from B5-G1. If it's one of their first few ranked games they're placing these people with ranked players that have the same ranked MMR that the unranked player has in normals MMR. It's the same elo they're used to playing at. It's not like you're S2 getting placed with a 0lp B5 scrub. you're getting placed with someone at your skill level who still hasn't played 10 games yet.
As for everything else, when you start ranked for the first time LoL uses your normals MMR to decide where to start your placements in ranked. After your first few placement matches it doesn't use your normal MMR anymore. What that means is that when you made a new account and started leveling it up you had a lot easier time winning games in early levels because you knew how to out play a level 5 player and so they started placing you with more difficult opponents earlier on than they did with your first account. I have a friend whose smurf was facing plats at level 27 because he's been hovering around G3/2 on his main and your MMR inflates a lot more when you have less games played.
The more games you play the more the system believes you're at your actual elo. Because of this it's not going to give you a ridiculous amount of LP for winning a few games in a row because since you've proved over a large scale of games that this is about where you fit in the system isn't going to believe that winning 2 or 3 in a row means you're necessarily better than your opponents.
Pick order is a random decision now so I doubt you're last pick in all of your promos but if you are and you're losing because you're last pick then you don't really deserve to climb. You should be able to fill and not feed in your games. If there's really a role that you aren't comfortable with for your promos and you're last pick politely list your preferences and the role that you feel you are least adept to play in ranked so that your team can work with you.
If you're consistently winning ranked games you will rise. I'm looking at your Lolking chart right now. It doesn't actually look like you've been that active recently as you have no data points between a month's span. Other than that looking at your match history you have almost a 70% winrate with Kat and you lost with her, you lost your first ranked game as Amumu, you lost your 50% winrate with Thresh that last lose. Those are your last 3 loses. You can't blame those on being 'last pick' (since I'm gonna assume that was a set of promos for you) if you're playing champions you do well on. I've played over 76% of my games as support in ranked, if I'm last pick and I get support I can't blame losing on getting last pick, In fact if you're lower on the pick ladder and you get your main it means you can either counterpick (if you're versed in the many champions of your lane) or at least avoid being hard countered (as the enemy team is more likely to have picked your lane opponent).
You'll get to Silver when you get to Silver. In fact you'll probably be placed in Silver 5 or 4 next season if you do well on your placements.