I'm not an expert at this, but this is my understanding of wave management.
Pushing is clearing the wave faster than your opponent.
Why should you push?
Generally it's for a purpose. You want to rush your level up and use the level advantage to get a free kill. Or you want to back, so you push the wave so that it will die to tower and reset in the middle.
Sometimes it's on accident. Tristana E passive naturally does extra damage around the one she kills, so she naturally will push the lane if both ADCs are just last hitting. If you trade into a minion wave, it will force their minions to focus you. If the enemy doesn't return damage, then that means your minions will continue to focus the enemy minions, thus cause the enemy minions to die first and the wave might push towards the enemies side of the map.
Look to see who's pushing. People get caught by level 2 so often because they are so focused on getting CS that they ignore that the enemy just leveled up. You need to match or be pushing early, if not, stand back so they can't kill you. Same with level 6.
Fast pushing is just clearing the wave as fast as possible while trying to get as much CS as you can.
Why should you fast push a wave?
Say you just got a kill but are low health and don't know where the enemy jungler is. Most of their minions are dead but there are still a few alive, and if you don't kill them, then the wave will be outside the enemy tower putting you at risk of being ganked. The enemy could also freeze it, which you don't want. So, here you'd blow your abilities on the wave just trying to clear it as fast as possible. If you get the CS, great, if not, it doesn't matter because you just want your minions to go under the tower and reset the wave back to the middle of the lane. You also want to do it quickly so their jungler doesn't just walk up and get a free kill since you're low health.
Another example is Baron is up but there is a wave bot lane threatening your open inhib. You don't have TP, so you can't really be spending a ton of time down there clearing it. So, do it quickly so you can get to Baron ASAP in case a fight breaks out.
Another example, as you pointed out, is that your enemy left lane and is seen in another lane. If you can follow (TP, TF ult, Shen ult), then you can easily follow them. But, if you can't, then the best thing you can do for your team is A) get tower plating/tower damage, B) roam to another lane to sorta balance out the pressure, C) follow and hope to clean up the fight. In every scenario, you want to push the wave first so that you aren't denying yourself, and you want to do it quickly so that the enemy laner/jungler doesn't punish you.
Another example is if it's early game and your jungler is invading or trying to do Dragon/Herald. If you can, push to get priority so that if something happens, you can follow without putting yourself behind. Obviously you can't always do this either due to your opponent being stronger, lacking mana/resource to push with, or due to being pushed in yourself, but ideally you want to push if your jungler is on your side of the map and might need your help.
Slow pushing is more complicated. What it basically means is letting your minions build up so you have more allied minions than enemy minions in the wave.
Why should you slow push?
Early game, you can slow push a wave to the enemy and then try to kill them. By doing so, they now miss this gigantic wave that will deny them EXP and gold. You'll also probably get a few tower plates with that many minions beating on a tower at once.
A slow push is also a great way to set up opportunities to back or roam. No one is going to leave 10+ minions to help an invade because they will be behind in EXP and gold, making it near impossible to fight in lane after that point. So, by slow pushing, you can then join your jungler in an invade where the enemy laner is forced to either help and lose EXP, or not help in which case you may get a free kill.
Later in the game though, you can use this is threaten tower without needing to be in the lane. Think of a super minion. Even with the enemy defending against it, the wave will almost never get out of their base. That's because the super minion is so strong that it wipes out the wave really fast. A large wave of normal minions can do the same thing. If you really want to see this, go into the practice tool and just watch the minion waves bounce back and forth. You'll start to notice how these really gigantic waves start forming and they will wipe out 3/4 of a towers health when they crash. That's a slow pushing wave.
If you know Baron is going to spawn in a bit, you can go to bot lane and set up a slow push. That way, while both teams are busy fighting over the Baron, your minions could be threatening their bot lane inhib, thus either forcing someone to not be at the Baron fight thus giving your team an advantage, or forcing them to possibly lose a tower because no one is defending.
How to set up a slow push?
It's a little more complicated than it was before. Minions now get buffs for the number of towers in a lane. So, if you have an open inhib and they have their outer still up, it will take more minions to counter their buffs. I don't have the exact numbers on this, so I can't say for certain what the numbers are.
Previously though, it was very easy to do late game. If you wanted to set up a slow push bot while you grouped at baron, you'd just kill the enemy caster minions and leave. Casters do the most damage, so by killing them, it means your wave shouldn't take much damage. When the next wave spawns, you'll have 4 or 5 old minions and your fresh 6 minions, while they'll have maybe 1 old one and 6 fresh ones. Most of your caster minions will be alive too, thus out DPSing the 3 that the enemy wave has, which means you'll slowly build up more and more casters as they push further down the lane, easily clearing any new wave until it hits either a champion or tower.
Freezing is keeping the wave in a specific spot. You do this by killing minions at the same pace as your opponent. Now, most people use the word freezing when they maintain the wave at their tower for more than a few minutes, so we'll use this definition.
What is a freeze?
Basically, it's an enemy slow push. You maintain more enemy minions than allied minions, so that as your newly spawned minions arrive, they are killed quickly so as to keep pushing to you. What makes it a freeze though is that you keep it in check. You trim off extra minions so that it only pushes as hard as you want it.
Why should you freeze?
Freezing is great if you are weak. Someone like Nasus might look to freeze since it allows him to get stacks easily without needing to be far out in lane, thus letting him get stronger without putting himself at risk. It's also great as an ADC if your support is roaming, because since you are 1v2, you want the safety of your tower to at least turn their dive into a 1 for 1 rather than just you dying.
Freezing is also great if you are strong. While most people want to push when ahead, sometimes it's best not to. If you push the wave to the enemy, then you are allowing them to get gold and exp while being safe under their tower. If you are strong enough, you might be able to tower dive them, but if you aren't, then you are just letting them scale for free. Even if they miss out on some CS they are still getting EXP. So, an alternative if you can't tower dive them is to freeze by your tower. Then, you deny them CS and possible deny them EXP, thus forcing them to either try to fight you, which you will probably win, or roam to another lane to try and recover. This is great against Nasus as it forces him to come out from under his tower where he can easily be ganked or even outright denied stacks. It's also great if the enemy support is roaming but you can't dive, as the enemy ADC is now forced to miss EXP and gold while his support isn't in the lane. Your support is then free to either roam as well, or just stand up in lane and ensure that the enemy ADC can't get in range.
How to set up a freeze?
Generally, you need to maintain 4 more enemy minions than however many allied minions you have. So, if you have 6 allied minions, you need 10 enemy minions in the wave. Casters are easily to maintain due to them being ranged. Melee's will often run under tower and break your freeze accidentally. A Cannon is stronger, so if the enemy has a cannon and your minions don't, then you'll need less to maintain the freeze.
One way to get this is to stop the enemy minions from hitting your tower by standing in their path. They will focus you and you will lose health, so you need to make sure no Champion is in range to punish you for doing this. Often best to do this when they are backing or out of the lane. If you can, kite the minions around. They will need to stop to walk, thus lowering the amount of damage you take.
Another way is if the enemy is slow pushing towards you. If you can maintain 4 more enemy minions than allied, then you can easily turn their slow push into a freeze. It might require a lot of trimming of the wave though, and if they are slow pushing, they probably won't just let you do it easily so you might need to trade a little to do it. Be careful though trading into a large wave as the minions can do a surprising amount of damage.
Note:
Any time you take an inhib, you are creating a freeze for the enemy essentially. In most cases, you will not get any CS from that lane for the duration of the inhib being down, thus you are losing out on EXP/Gold in exchange for pressure. If you don't do stuff with this pressure, then you are just hurting your team. You need to be taking Baron, Dragon, other inhibs, and/or other towers while their inhib is down. Otherwise, if you don't, you're just letting them catch up without needing to risk fighting you while you're strong.
Not properly utilizing pressure is the reason comebacks happen.
