Hi dmoney,
1) Yes it is possible that mail.franchise.com keeps a copy of your incoming email for backup, review, or other purposes, without leaving any trace of that activity in the mail headers.
2) Yes, even if not stored, it's still technically possible for the emails to have been seen for example by the system administrator if the emails were stuck in that mail queue for any length of time or by someone recording packets on the network. BTW #1 is the much more likely scenario.
3) Replies to emails would go out your outgoing mail server setting, or the Gmail servers if you are using the Gmail account for your replies. The emails sent out from Gmail wouldn't be seen by mail.franchise.com unless you were using a mail program on your computer to send the emails and it was configured to use that mail server.
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