[Android Apps and Games] mkasimd:
AFAIK, it works like this:

Login: OAuth

Sync inbox: IMAP

FairEmail by default uses IMAP for synchronizing the email inbox. It does not implement some type of a Gmail-API. Because let's be serious, even adding OAuth for each provider is a hassle in itself, but then also accessing Mails via am API varying for each provider would be just another hassle. Especially so if there is no publicly available API documentation. Google luckily has a proper Gmail API, but still.. FairEmail uses IMAP as far as I know. Maybe Gmail automatically allows IMAP access once authorized via OAuth?

I'm sure Marcel can clear this out once he's awake (be reminded it's basically 2 am in the Netherlands right now and he's the only developer as well as support member and maintainer for FairEmail)

Now to exporting mails. Open an email, press the three dots menu and export raw message. Save the Rae message. It is the raw message as defined in the RFCs and you will be able to open the file in any text editor (though base64 encoded content might look "unreadable" to you). You should also be able to open the file in Outlook or many other email apps. Because that's a standard email format as defined by the Internet Engineering Task Force.

FairEmail also supports showing the raw message as well as importing such raw messages into an inbox specified by you. I use that feature a lot when I receive a raw message some of my users want me to analyze because it has an abusive content or such. Because FairEmail shows the headers in a more accessible way. (Disclaimer: I manage the mail servers of the Liberal Democrats in Germany)

If those other email clients do export the emails formatted as an ARPA Internet Text Message or in the Internet Message Format as defined within the IETF (Internet Engineering Task Force), you should be able to import those messages into FairEmail.

Disabling the permanent notifications: see Marcel's answer to @Will_T just a few messages above. Go to the android notifications settings and select the channel or category "receive" and disable it. Leave 'nee messages" activated. It must be done in the Android notifications settings.

You cannot disable the watchdog permanently. You can only disable the notification from showing up. That has one simple reason: each Android OEM and flavor have their own battery optimizer and manager. Some stricter than the others. Strict ones close the app even if battery optimizations are disabled. Meaning users cannot receive notifications for incoming mails when that happens. So this is bypassed by a permanently active watchdog. That way, the battery optimization process thinks the app is still active and doesn't shut it down.

It sounds bad, but in practice it has no drawbacks in battery usage. I'm fact, if your server supports IMAP PUSH and MODSEQ, the battery usage should be much lower than with most other email apps, all while you receive incoming mails quickly within seconds.

Again, disclaimer: I am not a member of Marcel's team (cause he's a one man army anyways). So don't take my answer as credible as his. I am just answering according to my perhaps limited knowledge and with good faith and in hopes you get your matters resolved quickly as well as Marcel has one issue less on his plate when he wakes up. But he may still correct me later on if I happened to have said something wrong :)