I had my old personal website setup using the "Classic" google sites. I had a domain name (e.g. www.myname.net) that the old site mapped to (and .com and .org forwarded to). I transitioned my site to the "New Google Sites". It's published. I've followed their directions to use the custom url (so far as I can tell).
Currently, when I go to www.myname.net, if I am logged in, I get a redirect (not a map). If I am not logged in to my google account, I get redirected to login. It happens even if I use the non-custom url of https://sites.google.com/site/myname.
I've seen some conflicting information out there about whether the new sites can even map. But, even if it can't map, I should be able to get to the site through their url, without needing to login, right?
Specific Questions:
Any published new Google site is accessible via a URL of the form:
https://sites.google.com/view/my-site-name/home
So the answer to your first question is yes.
The answer to the second question is also yes, the new Google Sites can map to a custom URL. But it is not the most straightforwards thing in the world to achieve, as it (depending upon your starting point) may involve updating DNS entries, to then establish ownership of the domain with google, and finally then mapping the URL! And a further complication is that the process (currently) differs between a free/personal google account and a G-Suite account.
Also be aware that the process is not immediate, you may have to wait a few hours before you know it is working; so that may actually be your problem?
The Google Instructions:
Personal: https://support.google.com/sites/answer/9068867?hl=en
G-Suite: https://support.google.com/a/answer/7392423?hl=en
But I also found these links even more helpful:
https://www.steegle.com/google-sites/how-to/assign-custom-url-domain-personal
https://www.steegle.com/google-sites/how-to/map-domain-web-address-g-suite