Since you are going to use our DNS services for resolving the domains you would also need to add those domains to the DigitalOcean domain manager:

https://www.digitalocean.com/domains

Then it should work. =]

Check out: How To Create Vanity or Branded Nameservers with DigitalOcean Cloud Servers.

If you're going to use DigitalOcean's DNS service's IP addresses, you should add them to the DNS manager in your account control panel in order for the domain to resolve.

Hmm, I have tested in all the possible ways to solve this matter by myself. Here is the solution:

# If you want fully branded name server, then you need at least two NS. And for DNS my personal preference is either BIND-9 or PowerDNS, but it's up to you. Now, you can use a single droplet to configure two NS, but it might happen that (and it will happen!) at a certain time that droplet goes offline. So, all your sites will go offline too. So, you should use at least two different droplets.

# If you want to use your own NS record but not want to make things complicated, then you can use DO's IP and create your own NS record. For that, first create three Child NS or three Glue records using the three IPs which I have already mentioned in the original post. Now, create three A records just like what you have done in the previous steps. So,

ns1.mydomain.com => 198.199.120.125
ns2.mydomain.com => 141.0.170.89
ns3.mydomain.com => 198.199.95.114

in both A records and Child NS or Glue records.

Now, another A record like:
@.mydomain.com => [your droplet's IP]

Login to your DO a/c. From DNS, add a domain. There create these:

A record:
@.mydomain.com. => [your droplet IP]
ns1.mydomain.com. => 198.199.120.125
ns2.mydomain.com. => 141.0.170.89
ns3.mydomain.com. => 198.199.95.114

NS record:
ns1.mydomain.com.
ns2.mydomain.com.
ns3.mydomain.com.

Everything's finish now! Now, if you want to install cPanel, zpanel or whatever you want, you can add any new domain from your control panel. for newdomain.com, you will have to login in your domain's control panel (you do not need to add any further domain using DO's DNS!) and add:

A record:
@.mydomain.com. => [your droplet IP]

NS:
ns1.mydomain.com
ns2.mydomain.com
ns3.mydomain.com

Even if you create a new droplet, you don't need to add another domain for that droplet inside DO's DNS if you want to use cpanel etc. You just add the NS like newdomain.com, and one important thing, here in A record @.newdomain2.com => [your new droplet IP].

Even DO's new customers can use your NS from now on ;)

Have fun!