1) my laptop keeps discharging even when I close down correctly (the little white LED remains on).
I found the solution before on this forum which involved a Microsoft update and backdating a driver but I can't for the life of me find it.
2) the volume buttons on my keyboard - mute, volume down, volume up aren't working. Is this related to my previous problem at all?
For it remaining on, even with screen down, it could be if you have something like a mouse connected. Even if you put the laptop to sleep with the mouse plugged in, an accidental click can cause the laptop to wake. Especially if it is fairly quiet, you won't notice it and it'll drain battery as it is running, screen on, despite the lid being closed.
And yea, best thing I can recommend for the rest is to ensure all drivers are updated to specs on Asus' website (just search your laptop name along with "Asus drivers" and it should pop up).