Well first things first, the bottoms were both named nerd rage(or something similar) so either that was an insane coincidence or they were playing together.
Second, your top was solo against malphite(ouch) and jax(double ouch) until,
Third, you had a good jungler to help out.
Now onto you:
There isn't much to say. You got out farmed and harassed like nobodies business. now onto you and your partners first two deaths:
start around 9:45 watch the fight closely, although you are shifting focus a lot, when Leona throws her sword, she was aiming for Sona, but missed and hit Caitlyn, which I think caused both of your first deaths down there because had she hit Sona then you would have both been focusing on a weak Sona, and if Caitlyn had no support she probably would have fallen back. (but still you should feel bad that you only have 28 cs at the 10 minute mark).
Next start at 12 and watch as another simple mistake takes you down. Stop at 12:25 and see how your missed skill shot killed you both (that and your refusal to fall back or your inability to remember that Sona still exists). Had your Skill shot connect, no Sona. No Sona means no stun. No stun means you both focus a half life Caitlyn and melt her face off.
The things you need to work on right now are: if your going to have a skill shot character, make sure you an land the skill shots.
Make sure you always, always, always last hit. And don't forget to farm and harass at the same time. Watch as Caitlyn shots you and last shots creeps as opposed to your focus on one or the other.
Never lose track of you opponents, when you ignore Sona for about three seconds in your second engagement it got you killed (also I would have positioned myself a bit closer towards the turret and farther away from Leona)
Stop using your flash so damn much. Got to 10:10, if your flash was up you'd have been able to pop out of there no problem but instead... well instead you run at them guns blazing. When the fight is lost at least try to get away, unless you are positive you won't get away and your focus can kill someone.
Also if you have a good dedicated jungler, spam pins on your location when you're being harassed or pushed to hard, he'll probably check out the situation and come to help, and if not at least you tried.
Finally, and this is the least important point cause in solo games teamwork is always a **** shoot, your team fights are a disaster. The only thing you can do is tell people who to focus on (if you know) or ask who should be focused (if you don't). That way there is at least a vague Idea behind the madness.
All in all you didn't screw up to bad. Just keep practicing the basics (last hitting and zoning especially) and you should start to see improvements in no time, especially as your awareness starts to improve(cause in bot matches you almost always know were they are/are going to be as opposed to PVP where players run around like a crazy person).
PS it's a game, have fun and don't get upset too much (just enough to learn!).