Is it possible to send hundreds or thousands of email using, for example, Google or Yahoo's email server with an email address based on Godaddy.com?
Nope, its not possible. Though you can use google apps to send mail from gmail software using your own domain instead of gmail.com Check this for details: https://www.google.com/a/
Ok, will it be possible to send thousands of emails at the same time using another email software if I have this Google service?
Are you wanting to spoof the email address to make it look like it's coming from from an "@yahoo.com" or "@gmail.com" address but really isn't?
Me thinks and thinks, which I know can be a dangerous past time. But my first thoughts after reading your post is are you looking to use a fake email from a crediable company to spam?