From what you have explained, and provided I understand correctly there's actually a couple of answers.

If you use outlook to type and receive you email messages.

1.  Outlook is a stand-alone program, you can send directly through an ISP (internet service provider)  Or,, you maybe using Outlook and going through gmail? If so Gmail is just a Cgi bin to redirect email to and from your Outlook program.

2. You are using Gmail directly?  You sign in to gmail and use their editor (program, which is an online email program) to send receive email. In that case,  Outlook isn't involved with you sending email at all.

Clear as Mud so far ???  :-)   anyway

The extra email address is not an issue. The "coop address"  ISP provider may be configured to your Outlook program. If that's the case, IF you do use your Outlook program for email, the message would go through your ISP provider, Unless of course your Outlook program is configured to "talk to" Gmail. Gmail is then just a CGI bin.

Cgi bin's is like having a Post Office box, all your mail goes there. Then it is redirected to your "real" mailbox. ie: Outlook

You actually have 2 email programs. Gmail (on line based) and Outlook (in your computer)  They may or may not be tied together but, with out knowing which program you actually type messages in, I can't tell which way your email's set up.

Whatever you do,, Don't remove and/or disable your Outlook program. Outlook is intergrated into any/all other microsoft programs you run. M.S. programs being intergrated allows for lot's of really neat stuff to be done. and if Outlook is disabled/removed,, the other progams won't work correctly.

I hope I didn't leave you 3 miles back sitting in the middle of the dusty road wondering,,"HUH,, what in the HE!! did he just say?" 

At least you've got the benefit of my software degree for free.  I paid allot of money for the D @&%  Thing,, and has never made me one miserable nickle.

Just click on my name,, my email address is there, an I'll answer your question(s).