Is it possible for snow golems to "work" around a home? I'm talking about picking up items and opening doors, little stuff like that.
No, they just wander around aimlessly, leaving snow everywhere. They throw snowballs at hostile mobs, but it does no damage. The main uses for snowmen are amusement and luring hostile mobs away from you.
No, they can not do any of that, except what JLaBella said about the mobs.
But if you take supplies to make them into the End, Endermen won't go on the snow, so you basically have a safe zone.
Also, they throw snowballs at the Endermen, and it's pretty funny to watch. Hope this helps.
You could rig up some pressure plates to redstone contraptions and stick the snow golems in a pen with the plates; that would allow them to activate things on a psuedo-random schedule.