The problem is that incoming mails can have different formats, making it harder to process the content if you want to actually do something with it.

There have been drivers developed in the past that make connection via POP3 to fetch mails, process the content, and based on the content fill in parameters. So it is possible to do this, but the content needs to have a fixed format. I didn't test it yet with other technologies such as IMAP but it could be possible. Do note that GMail tends to be more strict on it on requires a verified e-mail client (such as Thunderbird), so it could be that it will be a bit harder to write your own client to connect to GMail if needed.

I'm not aware that the Generic Mail Receiver exists. If you want to have a "Generic" Mail Receiver then it should be possible, but your mails will then probably end in a table with columns "From", "To", "Subject", "Timestamp",... but the body of the mail will be harder to read as it can be formatted in HTML, or could contain images, or could also contain attachments -> then what to do with these?