Playing other champions really gives you a feel for how to handle situations. It is important to know what your champion is capapable of and what they are capable of. I can offer some tips though.
Watch your map: This one is the most important, always watch your map. When more than one person is missing on the enemy team you should be moving to a safe position. Make sure you know where your enemies are or where they were heading last seen. Keep the map scouted and when 3+ people are missing you should be trying to bunch up aswell. If theres 3 or more people missing and you're pushing a lane deep then most likely they are coming to kill you.
Know your role: If you're the tank, be the tank. Protect your squishies, stay close to them and protect them. They are usually your carries and thats how you will win the teamfights. Rally your team, don't be afraid to boss them around. "All push mid" "Group up". Late game power is all yours as the tank, you engage, you chose where and when to engage and where to push. Without you your team is pretty useless. If you're the carry, stick to the tank farm within moderation and keep an eye on the map. You're who they're disappearing to come kill. Stick to the brush when teamfights engage so they can't come in and just ice you and run away. Don't reveal yourself till some spells have been cast and they have already begun focusing another less squishy player. If you're support, keep the tank alive and watch your squishys.
Don't wander: If its late game and you're all level 18 its time to be a team, move together, stay close. Don't over extend and wandering also involves watching the maps. If the enemy team is pushed into their base and fending off super mionons go steal their camps, but just always farm in moderation and never over extend.
Know your enemy: Once again coming back to knowing the champions. If you're the tank and the majority of the enemy team damage dealers are ability power then stack magic resist, if you're a carry and a majority of the enemies are tanky high health grab a madreds and tear them down. Don't trust mobafire 100%. No build is gaurenteed every time. I say this all the time, LoL is dynamic and should be played as such. Build your build around the enemy team and what your team needs from you.
Also a side-note worth mentioning, once you can buy tier 3 runes build some pages for your favorite champions. Try to make ambiguous pages that can apply to for example all AD carries or all tanks. Some good purchase starting points are armor seals, armor pen marks, and magic pen marks. You can use them for basically any character.
The thing with teamrankeds is that the laning phase isn't the same as in soloqueue. In soloqueue the rotation isn't really there. While it would be good to have one, most of the time those 5 random people that are matched together just don't have enough teamplay to rotate right. In 5v5 Premades you can communicate better and you can try to get help from another lane. What I basically wanna say is: Focus on rotation right from the beginning and don't try the "I'll win my lane so I can carry" because most of the time you won't carry alone and as soon as you have an advantage you won't have to rotate too much. Also try to 3 or 4 man gank botlane a few times early because ADC is by far the most important role to get fed in premades.
So to answer your 3 questions:
How do I win against a push combo
Pusher comps usually have a lot of movementspeed, poke, splitpush or waveclear. They can siege very well and they can also take objectives down either because of high range or because of high objective-burst. Now what a pusher comp usually lacks is definitely the ability to teamfight. The best thing to do is Initiating on them as soon as someone who attacks the turret stands too close. I assume you named Caitlyn as an example because of her 650 range with autoattacks. But think about it: 650 Range may be huge but she still is vurnerable as soon as she starts attacking the turret. Also If a bruiser attacks the turret just ignore him and punish him for not protecting their squishy roles. Usually as soon as an ADC gets stunned he should be dead. Another thing that should be taken into concideration when playing against a pusher team is taking a champion with good waveclear. You actually have to take just one champion that can clear a wave fast and they can't really push (Cait won't risk a turret shot if there is a chance she gets stunned while having turret aggro).
How do I turn the game arond when the enemy team has the momentum?
Well this is actually pretty easy if they really push all lanes. Ward the jungle and go 5 man mid. Unless you haven't screwed up completely you will push a lot faster than they do and if you ward properly you can catch them as soon as they wanna stop you:
Example.
Shen pushing top
Their jungler is somewhere in top jungle
Supp/adc/mid pushing bot.
Now what you wanna do is Go mid as 5. Usually Botlane will stop pushing and come to you. Shen will wait till he can ult to the ADC/Midlaner. Since you know that they are coming from bot and that shen will ult you also know that their jungler isn't there. The jungler is usually rather tanky and in midgame teamfights tanks are the biggest problem. This means that you can go bot as 5 since there are 2-3 squishy targets with the potential of a tank teleporting in but as soon as one of the targets is dead there won't be enough damage to kill you 5 just because of a missing bruiser. Of course this isn't the only imaginable situation but I hope you get the point. Try to find somewone where you have a big chance of killing him.
Now if they aren't pushing all lanes and are sieging you have to clear waves again and try to find their mistakes. If they really are just as good as you and they have an advantage your only chance to win is by their mistakes (but don't worry they will do mistakes). You can try to make them feel overconfident by sending someone to a place that seems isolated while you are wating somewhere near. Another thing to try is if you have better scaling champions. If this is the case just try to stall.
How do we win with a poke combo?
Well there are two options as an addition to the poke part of your team:
Either you pick champions that can push well or:
You pick a champions with high mobility.
I really wouldn't recommend picking an AOE initiater or something like that since poke usually comes from the Midlaner/ADC and most of the champions that can poke well are rather poor teamfighters (sure there are exceptions but not too many). The main argument that speaks against picking an initiator is that you will lose against a team with a stronger teamfight midlane/ADC.
So as I already mentioned the best tactics here are: Hit&Run and Ultra siege.
Also try to make them feel unconfortable by baiting them to baron. This allows for poke and pushing potential (Just make sure that everything's warded) since you can't really take baron / dragon against a poke comp.
Now if you wanna know how to find good team comps just go for statistics (I made a small summary of the most common team comps here and what their strengths and weaknesses are if you're interested, but It's by far not everything you can say about teamrankeds). I assume that you know that guides won't help forever if you say that you play teamrankeds/tournaments. There are no guides but only statistics and small tips that will improve your playstyle after a certain level.
Best Answer
It takes time, and the self control not to kill all those minions ;) You'll find most answers in this Lol Forum topic
Long story short, you shall kill only the ennemy's mage minions first. They are the weakest, but they hit the hardest (twice as much as the melee minions if I remember correctly). Your minions will spend quite a bit of time to kill the solid melee minions, just enough time to let the next wave reach them, you'll then have 4-5 minions left in your first wave + the 6 (or 7) of the next wave. If the initial fight is in your part of the map, you may eventually kill the ranged minions of the opposite second wave, in order to build a hardcore minion wave, but it takes time.
Have fun.