As a jungler when should I push the lane after a gank?

To answer your original question:

Clearing waves is bad in some cases.

By just clearing the current wave, you are allowing the minions on your team to push closer to their turret. If their champion is capable of freezing the lane, this will put your buddy at a disadvantageous position if the wave you cleared was in the middle of the lane.

In general, you want to do one of the following if both are dead:

• Last hit the current wave or simply leave. (If the lane is neutral)
• Push the wave up to the tower and make the tower kill one or two of your waves. (This is to reset the lane, the closer minions are to pushing into an enemy tower, the more priority you should give this.)
• Clear part of a wave so they don't push into your own tower. (To prevent resetting the lane in the enemy's favour)

In other words, it depends on the situation, but a general rule is that you want to keep the lane as close to your side of the map as possible, and as the game drags on (if you are winning) you want to shift that balance from your side to their side more and more, so you can start putting pressure and forcing down turrets.

If the enemy is alive:

You either want to:

• Keep the lane so that you don't lose a turret as fast. (This is mostly true for strong pushing champs)*

• Leave the lane if the enemy is weak at pushing if you can do more beneficial stuff. (Get / give buffs, secure dragon, gank another lane, whatever)

• Kill the enemy champ. (If you think you can do so)

*Note: In some cases it is recommended to let your tower get destroyed, if the player who is laning against an enemy is performing really bad. Letting the tower get destroyed allows the minions to get pushed into your side of the map even further, allowing the bad laner to freeze his lane at the second turret and farm there. (Of course, you should pay attention at this point because otherwise you're just giving away turrets)

If teammate is alive but enemy is dead:

Either:

• Leave and let him continue farming.
• Help him push down the turret.*
• (Situational, mostly for bot-ganks) Kill Dragon

*: Pushing down turrets at botlane can be especially helpful as it allows you to pressure them more, thus giving you more leeway to take dragon out, as they can't permit to stay close to it if they don't maintain vision of the river and jungle 24/7.

Also, if there are other lanes that you can gank right away, you should of course do that instead of just farming minions for your buddy. A few minions aren't worth as much as another kill + assist.

And as a closing note: Ultimately it also sort of depends on the champion composition. Sometimes you have a team (or jungler) that is capable of pressuring a lot more than the enemy team, in which case again, you want to push up a bit more so you can take more control of the map, sooner.

When to push:

(Enemy laner is dead OR forced to back) AND (The lane will end up moving in the enemies direction OR Your teammate wishes to back quickly also)

This gives you 100% the time you want to push and this should be the ONLY time you ever tax.

Times you DO NOT want to push.

The lane can be frozen easily on your laners side AND your laner wishes to keep it frozen.

In the time to push, this part is the most important (Enemy laner is dead OR forced to back). If this is true, do not tax or push or your laner has every right to be pissed at you.

The second part of the AND clause is a little less clear: (The lane will end up moving in the enemies direction OR Your teammate wishes to back quickly also) Basically, if the wave is beginning a slow push towards the enemy turret, make it a fast push so it gets there before the enemy is back to lane. If it can be frozen and your laner wishes to keep it frozen, DO NOT TAX or push.

One key thing that throws out the rules. If an enemy is missing AND you are low health do not push.

So fully:

(Enemy laner is dead OR forced to back) AND (The lane will end up moving in the enemies direction OR Your teammate wishes to back quickly also) AND (You have high health OR no enemies are missing)