Can personally give my experience on both of these questions.
I would say I'm half and half for "Boss" type fates, and killing so many, escort, item turn in, destroy supplies, etc. fates that have given me Atma. My Western Thanalan Atma came off of a fate of nothing but killing coyote/dog/wolves. However a person in my party had received theirs from the Bubbly Bernie fate right before I got mine. Central Shroud I managed to pick mine up from a level 5 (maybe higher) fate that was literally right outside of the gates of Gridania (just south-east of it in fact). All I did there was kill raptors coming across a bridge. Central Thanalan I got it from the turn-ins with the Cactuar fate that requires you to pick up the cactus fruit.
Outer La Noscea I got it from one of the "Boss" type fates though, same with Upper, but West and Lower were just random kill stuff fates.
Also I have stayed in a zone long enough to do each fate roughly 3-4 times over, but at the same time I've been in a zone all of 5 minutes and gotten an Atma. So far I see no consistency in multiple completions, changing jobs, or teleporting around that have hindered my Atma collecting efforts in anyway. It truly is random.
At least from my perspective, your mileage may vary.