During my time at Fermilab working on an LHC sensor team, I forwarded my Fermilab email to my personal gmail account because their mail servers were so antiquated (100MB quota, no web interface, etc). Filters, multiple Inbox labs, and all my emails I responded to looked like they came from my Fermilab email address.
You can send as any email address you have access to. Some email clients show the GMail address as well when the email is sent through GMail's SMTP servers. It's suboptimal, but there are plenty of SMTP services you could use if you want to get around that.
I setup my email that way years ago and abandoned it. At that time, it caused a lot of my outgoing emails to get flagged as spam because of the mismatch of various mail headers between the Gmail domains with the various to/from/reply-to domains. Don't know if it would trip fewer spam filters now.
You can also configure it so that Gmail uses the other email address SMTP servers. However, I'm not quite sure whether they still include the originating Gmail address in the email headers.
