Setup DNS on Windows Server 2012 with DHCP running router connected to it

By default, Active Directory clients will update their own records using secure dynamic DNS updates. Once they are joined to the domain, they should immediately and automatically create this record in your ADI DNS zones.

It has nothing to do with DHCP in a default configuration. You can use DHCP proxy updates, but this is not recommended in most deployments and needs extra configuration. If you were using this configuration, you'd know it.

See this article for more info: http://technet.microsoft.com/en-us/library/cc771255.aspx

mDNS may interfere with you using the .local domain - http://en.m.wikipedia.org/wiki/.local

This is why its not recommended to make up your own domain names. .local is actually used.

Though windows doesn't have mDNS natively it often comes bundled with apple software. So check the PCs for zeroconf or similar software.

You may have to setup split horizon DNS. This is where external dns clients get sent the public IP but internal ones get sent local addresses.

Or fix the new router to work the way the old one did, allowing you to use the external IP internally. This may be configurable on the new router - its usually called nat loopback or hairpin nat.

Also, having a domain controller in the DMZ may not be the most secure way to do things...but that's a subject for a different question.