EDIT: Actually another discovery the other day -- NPC black phantoms don't invade if you're invaded by another player. So I had an invader somewhere and was trying to trigger Kirk in the Depths, but no go. When I killed the invader by Gaping Dragon's fog, Kirk invaded even though I wasn't in his area.
EDIT 2: Oh yeah, another. So after I beat 4 Kings I was able to warp out of the Abyss even though I didn't have the Lord Vessel yet. I guess that only makes sense, but I hadn't thought about it before.