When I try to use POP3 with SSL, Gmail complains: "Certificate is self-signed"

Oh lucky I went to sleep a lot later.

Oke first screenshot, very correct. You also will see that the namservers are still ns1.primarydomain.com and ns2.primarydomain.com if not this is correct anyway.

And then in Server Manager -> DNS Administration I currently have these 2 records:

Delete both. In Directadmin you don't set the rDNS/PTR, so you can loose the in-addr.arpa records.
The second is your www hostname, this is wrong so can also be deleted.

I'ts written in my manual to do it in DNS Administration. Not a DNS record, a complete DNS entry as if you would create a new domain. But not under a user account also not as admin in user level.

1. I don't need to create a new DNS record on the server for server.primarydomain.com ?

YES. In DNS Adminstration as admin in admin level. So there you enter server.primarydomain.com as new domain and your server ipv4 and then ns1.primarydomain.com and ns2.primarydomain.com and save.
Then you will have kind of the same as you now have for the same www record (which you will delete).
Then don't forget to via SSH in /etc/virtual create the server.primarydomain.com directory and chown it to mail:mail.

2. I assume this [backwards ip address].in-addr.arpa record is my Reverse DNS record.

Officially they are reverse dns records. But they won't work if you create them in Directadmin in this case, you need to set the ptr/rdns record at your server provider. So remove all those in-addr.arpa records, you don't need them.

c) And then I also need to contact the owner of my IP address and get them to add my rDNS/PTR records.

Correct, that is where the rDNS/PTR is made for both ipv4 and ipv6.

And then lastly, there's CloudFlare. This is what I currently have...

I don't use Cloudflare, so I don't know how this is working if you are using your own nameservers like ns1.primarydomain.com and ns2.primarydomain.com.
If you only use cloudlfare, there is probably not much you have to change. You need to add the server A record there maybe.
If you use cloudflare as external DNS, I would suggest for primarydomain to copy also the SPF and DKIM TXT records.
But as said, I don't use cloudflare, so if Cloudflare just works as external dns then it should be like I said.

Maybe @Zhenyapan or @jamgames2 work with cloudflare and can confirm or tell you which records you need additionally in Cloudflare.

I really have to go offline now it's 02.33 already.