1. Where can I see the BackUp Gmail errors?
2. Get “Error logging into account” error but my credentials are right.
3. How are my emails saved?
4. Get “AllMail folder not found” error.

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1. Where can I see the BackUp Gmail errors?

The App writes the messages in the logs and you can read them in the Console.

Open it (Application -> Utility -> Console) and look for the BackUp Gmail lines:

2. Get “Error logging into account” error but my credentials are right.

You need to enable the IMAP access in your Gmail settings.
If you not resolve maybe you have the 2-step verification (verification code) enabled, create a new Application-specific passwords for BackUp Gmail App and use it.

3. How are my emails saved?

An archive created in the directory that you choose under the Preferences menu.  In this directory you will find a db.sqlite file and a FileStore folder.  The database stores all relevant data from every single email (ld, data, folders, etc).  In the FileStore folder, the emails are saved in the .eml format.

4. Get “AllMail folder not found” error.

You need to enable the “AllMail” folder in your Gmail settings.