#1
Is there a proper way to position the pickup relative to the strings or does it vary entirely? I'm trying to make sure my humbucker isn't too far or close to the strings in order to get the most out of the guitar.
#2
The only rule is this: Play around with it until you get a sound you LIKE.

Now, with passive pickups, if you get the pup too close to the strings, the magnet might start to influence your sustain in a bad way, so be aware of that.

With active pups, a lot of people say to put them as close to the strings as possible, since they don't have a very strong magnetic field. What my ears tell me (with my SD Blackouts, at least) is that they sound better when adjusted further away; about 4-5 mm of space between them and the strings. To me, they sound less "processed," have better touch dynamics, and essentially sound a little warmer and more like a high-output passive pickup.
#4
To add onto what he said, I also heard they become "hotter" in tone and more prominent the closer they are to the strings as well. So try not to over do it, but find the sweet spot you like.
#5
True, standard magnetic pickups do get hotter as you bring them closer to the strings, too close and they decrease sustain and can pull strings out of tune too. I've also had some that will pick up noise when I hit the pickups while playing if too close. Putting them closer to the strings can also make them sound dirtier in some cases.

Tinker with it and see what you like, make sure all are the same volume level when you get the basic height you like, just let a chord ring and switch back and forth between pickups. You should hear no change in volume level.

You can also fine tune them for tone, by putting one end closer to the strings than the other. I do that a lot with neck pickups to get better treble response without the bass getting too heavy. I did the opposite with my Strat bridge pickup to bring out the bass and not get too bright. Moved the treble end lower and the bass end higher...
#6
It varies a lot from pup to pup. I was severely dissapointed with a Seymour Duncan Hot P90 when I first put it into one of my teles. I set it to where I generally run passive pups and it sounded weak as piss. After adjusting it, it came to life. It just needed to be run closer to the strings - and we're only talking about a couple of mm here. Experiment, man.