I like the theory that Azra is a place that actually exists. I'm thinking that Azra, the city of the future, is a superior civilization because humans were able to discover ways to manipulate genetics to a point where they could give themselves powers. The whole city is an experiment that promised breakthroughs in human development funded through governments around the world.
The "gift" or whatever it is was the result of genetic manipulation. This society started using these genetic manipulations to give people enhanced abilities. There could be a couple of scenarios(they aren't mutually exclusive to each other):
(1)The wars/collapse of society started because government militaries around the world found out how to copy this process for soldiers and they accidentally created imperfect/dark beings which couldn't be controlled.
(2) Azra went to war with the rest of the world either because they were attacked by other countries OR they themselves were the instigators, which caused the collapse of society, and they ended up winning.
(3) Azra loaned genetically modified people to other countries for militaries and other tasks which required enhanced humans, but they relapsed into going "dark" over time which caused wars and the collapse.
(4) Azra is gone because they collapsed under their own creations. Think of it like the matrix.
The badlands originally was a walled off territory (and probably isn't the only one) created by whatever government/military in order to escape the fallout. In time, that political structure collapsed too. Barons came about as a means to keep people in check.
As far as the monastery, it's just rouge people who escaped Azra because they didn't want to participate in the wars or escaped Azra due to whatever reason, and also wanted to learn how to control the power. We don't know how long it's been there but I'm going to assume a good amount of time. In order to have the gift, somebody in your family would have to have had it. We know it can be "taken away" but more than likely there's probably a way to inject it into someone(in the case of the people who first invented it) and there's likely a way to reactivate it if you've already had it(in the case of MK and the Widow who wants it back).
Bajie clearly knows a lot more than any other character, besides the master, about Azra and what it actually is. The fact he was able to reach out to Azra from that communications tower means that something or someone could be listening. It'll be interesting to see where the series goes with season 3. Please feel free to criticize my theories. I like these discussions :)