“Field StageName does not exist” when creating a formula for Case Object

You do not have direct access to the ownership change page so there's no direct way of doing this. I can think of two option and both these options will work only if the sidebar is enabled in your org:

1) Create a custom VF page which will mimic the standard change owner page. As this is a custom page you will have full control to check / uncheck the checkbox based on your condition. Create a home page sidebar component which will contain a script (jQuery / javascript) which will override the [change Owner] link to you VF page. So whenever a user click on the [change owner] link, he will be redirected to your VF page.

2) Now, If you do not want to create a VF page then create a home page component which will contain a script, this script will run only when you click on the change owner link, the script will check if the case record record type and the owner that is being changed to and based on this it will check / uncheck the "send email notification" checkbox.

Rather than creating "dummy" record types, which seems like a heavy handed way to identify the different incoming Email2case types, why don't you add different case origins to the picklist, and set three of these?

If you navigate to

Customize > Cases > Fields > Case Origin

you could then add three new values to the picklist

Refund X Refund XY Refund XYZ

And then in your Email2case setup (which you seem well familiar with) set the different Case Origin fields for the three e-mail addresses. You would then go ahead with the workflow rules you outlined (so again, I assume familiarity) to detect a new case being created with a Case.Origin of "Refund X" and do a field update to set the Product__c field to what you want?

If case origin being "Email" is then important to you still, you can of course add a second field update to set that back to a traditional value?

It is basically still using your approach of seperating the sources of the Emails, but Case Origin seems like a more appropriate (it's in the name!) way of doing this.