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You asked about something similar to the Chrome add-on which would work with Firefox. The Emailpet service described in that link (which costs US $5/month or US $48/year) connects to your account remotely (via IMAP, I assume) and performs the indicated features in the background. The setup shown at that link is make in the Emailpet application.
The Gmail filters can't perform forwarding on outgoing messages. Some other email services (such as Fastmail) can be set up for automatic BCC on outgoing messages.
Bill
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