So upon completion of the quest you will get a series of "System" messages that pop up directly after you talk to Jalzahn. They are as follows:
System: By participating in certain FATEs while equipped with a relic weapon zenith, you may obtain an atma crystal. The locations of the FATEs and the atma yielded are as follows:
Gives the whole list of Zones and the name of the Atma obtained from said zone.
System: This information can be reviewed in the completed journal for the quest “Up in Arms,” which can be found under “Side Story Quests.”
Word for word, if you want to confirm this you can pick up the quest yourself or go here: http://xivdb.com/?quest/66971/Up-in-Arms
The problem with your above method is the fact you would need almost everyone currently doing the quest to log their data of how long it took them to get their Atma, and from what series of Fates they got it from. Not to mention the level of the fate, if it had a previous Fate that lead up to it, or was spawned due to a "Failed" fate. The only thing anyone can tell you is that the drop rate is rare, and will take you some time to finish the whole quest.
The alternative would be having 1 person sit in 1 particular zone with a Zenith Weapon equipped, collecting multiples of the Atma that drop in that zone from fates, and collecting data from that. However I doubt anyone has the time or patience to take on this task, even if they had all classes with Zenith Weapons there's only so much fate grinding you can do before you go crazy.
From personal experience I haven't had much luck with the drop rate, my shortest wait time was roughly 2 hours, and so far my highest has been around 4 maybe 5. I can tell you though, my 3rd Atma I obtained was literally from a level 10 (maybe lower) fate in North Shroud that was close to Gridania. However North Shroud has a huge level range for the amount of fates it can hold, ranging from level 3 to 50. I currently only have 3 total Atma, so I'm no expert in the drop rates, just giving my experience.
I also completely agree, these are great points to get hard evidence on, just your second request is a lot harder to obtain information for.