If {Viridian Longbow} is equipped to, say, {Typhoid Rats}, if you tap the equipped rat to do 1 damage (using the ability on the artifact) to a 6/6 does it instakill it because of deathtouch?
702.2c (http://imtgapp.com/forum/index.php?action=imtg;area=rule;number=702.2c): A creature with toughness greater than 0 that's been dealt damage by a source with deathtouch since the last time state-based actions were checked is destroyed as a state-based action. See rule 704.
Yes.
It's all based on what the card says. In this case the equipment is giving the creature the ability to shoot the damage. Since the creature has deathtouch as well, the creature it hits will die.