Configured Airport as Bridge and use OS X Server for DHCP and DNS, no Internet
You're trying to accomplish more in one step than you can handle. Break down your problem into subproblems, and solve each one.

The first you want to get working is to have an internet connection on your Mac Mini when the Airport is set in bridge mode. Do this without setting up DNS-servers and changing DNS settings to use your Mac Mini as the DNS server.

When you have working, you can add on the DNS.

Basically at this point you want to ensure:

a) That the Mac Mini's network port is setup with the correct IP, subnet mask and default gateway. This is the static IP and network settings, you have been supplied by your ISP.

b) That your Mac Mini can "ping" addresses on the internet. Open Network Utility and try to an address on the Internet. Try first with an IP (such as 23.78.125.15) and then with a DNS-name (such as www.apple.com).

Remember that in this (simple) scenario, you can only use an IP-address once. I.e. if you assign a static IP to the AirPort Extreme, you cannot assign the same static IP to your Mac Mini.

If you have got only one static IP, you cannot use that IP on both the Airport Extreme and on your Mac Mini.