To verify DNS, launch Terminal.app from Applications > Utilities and issue the harmless diagnostic command:
sudo changeip -checkhostname
That'll display some host information, then an indication that no changes to DNS or related are required, or that there are DNS or related errors.
The server name as the domain name in a default install is a longstanding and expected behavior of OS X Server, if you don't already have DNS running. It's a little odd-looking, but it does behave entirely correctly for a local host.
Now the next thing I do with that is clean that out and set up DNS services for the local network, but that's fodder for another discussion. What the installer does is good enough to get OS X Server installed and going.
As for the OP, I would recommend not using .private or any other made-up domains. Spend the ~US$10 per year and get yourself a real and registered domain, or use a subdomain of some domain you already have registered. (FWIW and AFAIK, .private isn't an officially-registered top-level "private" domain, and all sorts of new top-level domains are coming online and making me rather skittish about recommending that and other "bogus" domains. Not until there's a registered "private" domain, that is. Or just use a real and registered domain.)