The tooki seems to spawn every 5-10 minutes. However as Nucleon said the drop rate seems to be quite low; I've killed about 20 so far with no luck. However I can confirm you can hop between channels to kill different ones.
No, it has 2 spawn points and respawn to the other one instantly when killed.
The drop rate do is quite low tho, I spend an hour alternating between the 2 spawn points
Got mine after 40-50 kills. Its not hard, you just can't take breaks. You kill lost tooki at one location, triport close to the other location, run, kill the lost tooki at that location, go back, repeat. All told, about 2 kills/5 minutes.
Took me 31 kills. You have to kill him in one spot and then go to the another (Up and Down). Or just with a friend or party one kill down and the other up, that's more quickly
Here's a tip for anyone doing this who doesn't have a friend available who needs it as well. Instead of moving back and forth between the two spawn locations, use two characters and just keep switching back and forth between them. It's MUCH faster than triporting and riding over with the same character, unless you have a PC that loads really slow on character swap.