As to issue 1: Are you clearing cookies on exit from Chrome? [And I don't care whether that's via an extension, an option in Chrome for doing so, or by running something that clears them such as CCleaner, BleachBit, or System Ninja]. Staying logged in, and not just to Gmail, is dependent on the retention of cookies between browser sessions.
As to issue 2: Have you cleared any other browser data, besides cookies? The memory of previously used addresses is a part of the broader browser history function, though at the moment I can't remember which specific data store (though it's not cookies).
Given that both behaviors hinge on things Chrome would typically keep stored, unless removed intentionally, I'm suspecting some sort of cleaning has been done.
Brian - Windows 10 Pro, 64-Bit, Version 2004, Build 19041
The purpose of education is not to validate ignorance but to overcome it.