Disclaimer: I'm constantly in the process of learning. Some stuff is my personal views. I may also get things wrong or misinterpreted. Let me know if I fuck up. Proceed with caution, thanks.
What are Incarnate Angels?
Incarnate angel is a descriptive phrase for souls that are angel in origin living in a human body from birth.
Angelkin is a sub-category of otherkin. It is used to describe people who identify as an angel. It encompasses a wider variety of experiences that aren't necessarily spiritual based (psychological, ficitional, etc.).
Do you have to be a part of a particular religion to be an angel/interact with them?
Nope. You can do whatever you want, believe whatever you want, worship whomever you want, such is the glory of free will. You may feel more predisposed towards a religion, world-view, or lack thereof which can be influenced your soul, but honestly follow your heart.
Angels also generally don't care what background you come from, just be aware of religious appropriation when it comes to angel-related information online.
Do angels incarnate because they are Fallen (cast out)?
No, but Fallen angels can incarnate too, so sometimes.
Variety of reasons. It can be summed up to about three VERY GENERAL reasons as far as I'm aware, and each one can be a lot more detailed than the following very general descriptions. Possible reasons are: 1) To help, 2) To live, and 3) To learn. For example, I'm mainly here to just experience what it is to be human.
How do you discover if you're an incarnate angel?
The specifics defer from person to person. Some learn/remember very quickly, many others spend years going back and forth. Generally, a lot of introspection is required because ultimately it's you and only you that can say who you are.
There is no requirement/expectation for what an incarnate angel looks like, behaves like, thinks like, etc. either.
It's best to approach it the same as other questioning otherkin. Otherkin.net and BeyondHumanity have a good FAQ/how-to.
Technically, you can search up stuff online specifically about incarnate angels, but I am very serious when I say the New Age movement (which is where you'll find 99.999% of this information) is heavily drenched in white supremacy. Remember Garfield, you are not immune to propaganda.
I highly recommend creature-wizard's blog for information about the New Age movement.