In Android contexts, should the 'a' in 'app' be uppercase always or lowercase?
Lowercase, except at the beginning of a sentence. App is short for application. And, given that you wouldn't capitalize application, no need to capitalize app.
In a formal paper, you should consider writing out application. At least for the first usage like this: application (app).
Here is a search of the word app on Engadget. Note the consistent lowercase usage. (Click to enlarge:)