The issue is not related to the name servers, but to the actual DNS records in the Cloudflare Dashboard.
The certificate, from what you tell me seems installed correctly. There shouldn’t be other things. You could try pausing Coudflare (it will expose your origin, but it would be for a little bit just for me to check some things).
You need to enable Always Use HTTPS and possibly Automatic HTTPS Rewrites.
Again, as above. The www version is due to the fact that there isn’t a DNS record (duplicate the record for www or add a CNAME, as mentioned above), both have issues with the origin not accepting things. It’s not Cloudflare.
PS: reply to my posts, not the thread, I don’t get notifications otherwise.