New domain name registered with Google - can it be transferred?

Long story, but bear with me:

I currently have peppermintvault.com here at Winhost, because the service is exceptional and I haven't had an issue in years (awesome job, everyone).

However, I also haven't used the "web side" portion in a while, primarily using it for email.

I didn't think it was necessary to spend the money for "complete hosting" when I just wanted email, so I've disabled auto-renew on my domain name and was looking into email services for my new domain.

I just tried to register for Office and was just told they do not use Google domains and that I had to register through GoDaddy (which I will never, ever support - these [censored to be family friendly] are the reason I'm using this service!). I'm now in the midst of a second refund attempt as another email service didn't disclose it would cost additional money to set up multiple email accounts (under the same domain!).

This has been frustrating, so I've decided to stop looking and will stick with Winhost.

Now, my situation: my SITE renewal doesn't end until 12/19, but my domain (peppermintvault) expires next month (which I do NOT want to renew).

I only registered through Google thinking it was no different than any other domain registrant, but I'm getting the feeling this may not be the case.

So, before I go spending more money (which ends up as a refund), I need to know:
-Can I use the newly purchased domain through Google at Winhost?

-Can I change the SITE name to the new domain name? (even if I don't plan on using the web site, I would still add a "this is [domain] site" front page.

I'm trying to determine if it'll be easier to set up a new site with the new domain or "flip" the existing site I already have.

I'm not opposed to letting the year go if it's easier to set up a new site.

I just want an email server I can control.

Thanks, and sorry for the long read