There are several answers to this question: But yes is the overall response.
1) A server with a dynamic IP address is NEVER a good idea. The headaches you will encounter as a result of this will not be worth the effort to make this work. An experienced SA could make this work, but if you're asking this question: You definitely don't have the experience or subject matter knowledge to make it work from the git go.
2) You actually do have a static IP. You have a buttload of them actually.
Since you stated you have an Airport Express (or most routers for that matter) you have the ability to setup static IP addresses in the private IP space (192.168, 172.16 and the 10.0. addressing) in the setup of the Airport Express under Internet, DHCP, DHCP Reservations. Enter an IP address that you will use on the Mac Mini Server during the setup. For example, if your DHCP addressing is, say 192.168.1.1 to 192.168.1.50, enter 192.168.1.1 in the DHCP reservation section and then enter the MAC address of the server. Once a static assignment has been made, you either let the server get it's IP form the AE using DHCP and it will get the static IP based on its MAC address, or just assign the IP in the Network Setup util in the Mac Mini (the latter being the easiest and less problematic.)
(There's MAC addressing, port forwarding, and NAT issues involved as well depending on how you set up your Mac Mini Server, but that is well beyond the scope of my simple reply.)
The Airport Express in your scenario can do a lot of the heavy work in your network setup.
Basically, if I am reading your question correctly -
Your AE sits between your internet connection and your MacBook Pros. Your AE is assigned a dynamic IP from your ISP. Then as your MacBook Pros connect to the AE, it in itself is handing out dynamic private IPs for the MBP's to use to connect to the internet.
All that is needed, is the static assignment of the server and point the MBPs to that server address and wahlah you have your file storage, access, serving, etc.
Hope this helps,
Scott