I've spotted Endermen far off in the distance, and they haven't appeared to notice my gaze. Within what range will they react to me? Does draw distance affect this range in any way?
According to the wiki, the range is 64 meters.
They are generally neutral. However, if a player looks at them by placing his or her crosshair directly over them for half a second, they freeze, stare back at the player, open their mouth and become hostile. This only happens, however, if the player is not farther than 64 meters from them. They stay frozen as long as the player doesn't move the mouse at all, even if just one pixel.
I spent a night on top of a town church and antagonised all the Endermen I could see. I could get them to look at me no matter how far away they spawned, though getting the crosshair exactly over them at extreme distances was actually hard and took a few tries to get their attention.
Though it's not an exact number obviously, I can attest that Endermen will notice you looking at them from several chunks away.
All AI have a range of 16 blocks.. The two exceptions to that are Ghasts and Endermen.
As far as I can tell,
if you can see it (its rendering) then you can anger it the 64 block range on the wiki seems fairly accurate. It will teleport approx. 16 blocks per hop, and it can not seem to teleport vertically (My testing area is on a structure atop a floating rock).
So, I will post a new video hopefully tonight, but every time I tried to get a new video, it started raining on me or I would spend an entire night and not have a single enderman spawn.
Here is what I have found that is muddying up this discussion however. You do not stare At the body of the enderman to anger it.. You actually need to stare a bit over its head. This is less apparent at farther ranges but ended up being what I found out last night then had verified by other sources.
EDIT: Aside from the striking out section, here is the video as promised: