I also send myself e-mails to keep them safe and for future reference, but I rather send FROM Gmail TO another email system like Inbox.com or Outlook.com
That way if something happens to either one of them, I will still have access to the other.
Make sure you use different passwords.
In gmail a message is stored only once, it does not have folders. What you see as folders are filters with each filter displaying tose email that comply with the filter's rules. deleting a message will classify it as deleted and it will appear in bin
i had a similar challenge where my DVR was setup to send me screen captures when motion is detected on the camera's (between certain times). using my gmail account to send to my gmail account resulted in the same challenge. labelling/filtering the emails was a mission. I ended up falling back to my yahoo account (which i havent used in 15 years ), to send to gmail.