Is it possible to have ALL the audio (console and gamepad) output through the gamepad?

if you meant if you get the sound from both the TV and gamepad, then yeah, I think so

Depends on what you use the Gamepad for. If you're referring to all the time no matter what, the answer is no.

I am referring to all the time. I'm trying to set up my WiiU so that I can use it on my PC monitor. The video works fine, but when I plug headphones into the gamepad, all I get is sound from the gamepad.

Depends on the game. Some games put all the audio through the GamePad, some put parts of it, some put none and some put none through until you plug in headphones. With the games I have Assassin's Creed 3 and New SMBU work fine for this and AC3 is all sorts of well done on this. Nano Assault does an alright job but you lose some of the less important audio... or at least as far as I can tell. Nintendo Land just plan old fails although it doesn't matter too much.

Nope you can't unfortunately

I suggested this to Nintendo as an update after my Wii U froze while playing Nintendo Land(froze 3 times in a row, never again though), as well as others. They might, but probably won't. Only a few games have the option to have ALL audio go through the gamepad, like ZombiU, or COD(using only Gamepad). I gave up on that as a main source for audio for what would usually be on the TV, after realizing this... Your TV/reciever probably has a headphone jack.