Actualy, It's a couple of questions:
In the Google Cloud console, go to the Custom Domains tab of the App Engine Settings page. Click Add a custom domain. If your domain is already verified, the domain appears in the Select the domain you want to use section. Select the domain from the drop-down menu and click Continue.
You can use a custom domain for a site published on new Google Sites. Using a custom domain can make it easier for people to find and remember your site with a web address like www.yourdomain.com. To use a custom domain for your site, you can: Select a domain from your Google Domains account (if you have one).
Google Workspace essentially lets you create an entire Google account with your own custom email, instead of using a Gmail address. This not only gives you access to Gmail with your own domain name, but it also gives you separate access to other Google tools such as: Drive. Docs.
Here is a couple of answers:
Edit: The 3rd answer has been updated with the input provided by Nick Johnson in a comment.
With the the (new?) developer console it is possible add domains to apps for free without using Google Apps. But you will not be able to use HTTPS without registering the domain with Google Apps.
So the answer to the first question is "Yes" (presuming TLS is not required). The second question is not relevant. And as for the third question, multiple domains can be linked with the same application (without using Google Apps).
I can't currently find Google-blessed documentation to support this. But I am currently serving an App Engine application on a domain that is not registered with Google Apps.
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