Setting up DNS

This would be a very bad idea. Save yourself a lot of trouble and don't go down this road.

Most of your devices on the network are going to cache DNS info. If your sever changes IP those devices won't be able to find it (at least for a while). If you want something to resolve to a URL, you need to be using a fixed IP (either public or on your LAN).

You can't. a domain name requires the IP address to not change, otherwise the domain name would very likely end up pointing to an incorrect address.

It makes no sense to want a domain name on changing address.

Use a static address, outside of the range of the DHCP, and then assign the domain name to that.

What you are describing and requesting is Dynamic DNS. Apple unfortunately don't support this, the underlying DNS server software Apple use which is called 'named' aka. 'BIND' does support this but in order for this to work you also need a DHCP server that can be integrated with the DNS server so that when the client gets a new IP address via DHCP it informs the DNS server so the DNS server can update the record to match.

Apple's DHCP server definitely does not support this.

Note: In theory you can manually configure BIND to do its half of this, however this would all have to be done by hand not via Apple's front-end. Frankly not worth the grief.

Microsoft's DNS and DHCP servers do support this and this is actually needed for Active Directory to work properly.