I'll address only one point.
> 1. How is Google Push related to IMAP IDLE?
First, "google push" is not a term I often see. I assume you mean
"using GCM messages, now called FCM, to cause an app to begin
processing", with the usual scheme being that somehow a FCM message is
sent from a server via the FCM cloud and to the phone, which causes it
to do something like check in with the server.
First, this solution is problematic from a privacy perspective (and
requires agreeing to some Google TOS), and relies on having google code
in the phone. It is, for example, not present in LineageOS.
I am not aware of any use of FCM in k-9. (Perhaps that's because I use
the f-droid build, where any FCM code would have to be removed in order
to produce a Free Software apk. But my I have also not seen any
discussion of it.)
I am not aware of IETF standards that describe how to integrate IMAP and
these sorts of push notification services. There is a generic http
push RFC:
https://tools.ietf.org/html/rfc8030
and some more or less expired content:
https://tools.ietf.org/html/rfc4550
https://datatracker.ietf.org/doc/draft-maes-lemonade-p-imap/
but I am not aware of this being in use in a general/interoperable
manner. (It would not surprise me that various proprietary mail
services with proprietary companion apps do things like this.)
So I will advance the theory that:
k-9 does not use FCM/GCM (and hence does not use "google push")
IMAP IDLE is the standard approach
discussion of "google push"/FCM in this context is without basis and
basically a red herring
I would be interested to hear any explanations if that's not accurate.