edit - just read your update.
My guess is that it's just your Hotmail forwarding. Can't you get hotmail to scan for mail, then check if the date/time matches in gmail?
Or if in doubt, disable the hotmail forwarding for a few days and check if the logs stop in gmail. Then re-enable your forwarding, and check again?
/double edit - the texts are likely for any applications that need to access you email (like the hotmail forwarding).
I had issues with someone logging into my gmail account several months back and since using the 2-step verification, I've had no issues.