Well first let me explain how ranked works. I know you probably didn't intend to write it this way, but when you lose, and are deranked, you do not go from silver 3, to silver 2. Silver 2 is ahead of silver 3. I'm sure that was just a error on your part there.
I climbed to gold 4 in just a few days( i also just started playing ranked for the fun of it). I placed Silver 2 from the placements, then easily breezed my way to where i am now, but i am currently not playing ranked till some champion bugs get sorted out. A lot of my main champs have serious game altering bugs right now, and i won't lose LP because of Riots laziness. EX Syndra, Vi, Jhin, etc etc.
Anyway, it really doesn't "Get better" so to speak. Every tier has it's feeders, it's trolls, it's laggers, and it's AFKers, as well as little children who fight eachother more than the enemy. Iv'e actually gotten more trolls in gold, than i did in silver. Which is funny to me, you would expect the players to be better, and maybe more mature, but that's just not how it works sadly. I mean my first match, MY VERY FIRST MATCH, in gold 5, my laner literally inted. I mean hard core, ran under my turret, sat there, and died over and over all match. And nobody knows why...
What i suggest is this, when you go on loss streaks, take a break (like you said you did), and come back the next day with a cool head. At least then you won't be tilted from a huge loss streak. Also, never lose that many times in a row man. I call it at 3 losses. 10 is ridiculous.
Focus on what you can improve, CS, roaming, ganking, objectives, team fights, whatever it may be, focus on something you could of done better, WIN OR LOSS, always look for something you can do better. Like i said, you can't control your team, but you can control how good you get.
Also, concerning the "free win", that will happen the next time you reach 100 LP, you will be granted 1 free win out of your promos, there for you only have to win one more match to be promoted.
Play the champs you are comfortable and good at. Don't pick a "Carry champ" that you have never played just because it's a strong champ.
As long as you de-tilt yourself, focus on what you can improve, and keep grinding, eventually, theoretically, you should climb. Climbing really isn't necessarily up to your skill, as much as it's up to how well your team is able to cooperate and work together. It just takes a lot of time, and a lot of flips of the coin.
GL my friend.
