Q: What is the basic requirements to make a server work.

First off did you purchase OS X Lion Server from the App Store and are you using the Server Admin tools to confgure your services? From reading your posts it reads like you are running standard Lion and turning on personal file sharing and screen sharing.

Prior to Lion Server...OS X Server software was a completely separate install. With Lion you purchase the add-on to convert any machine running Lion into Lion Server. Easy and simple...alas a little dumbed down compared to previous versions of Server software.

Verify this and I will try and help you out.

So you are running a Mac Mini configured with OS X Lion Server. Internal use only. You are running the Server app to configure your server.

You want to provide what type of services to your intranet/lan clients? Are you in a mixed Windows/OS X environment? Are you wanting to do File Sharing? Address Book? iCal? Wikis? User/Group Profile management? Web Server? Email Server?

Explain your scenerio and your concern a little more so I can understand better.

Assuming your network is configured within a private IP address space (preferably 10.0.0.0/8 and 172.16.0.0/12 subnets and outside 192.168.0.0/16, as specific parts within the latter will mess up VPN services) and if you don't have another DNS server within your private address space, then here is how to establish LAN DNS services on Mac OS X Server.  (There are links there to other network-related information.)

The "private" stuff works, as stated in the article.  The "fun" here is for two reasons: that many sites will eventually decide to and start using their domain names more publicly (and $10 per year for a real and registered domain is cheap insurance against rebuilding the whole network with a new domain) and that that the ICANN folks have recently opened up the top-level domain (TLD) registrations to pretty much any TLD that anybody that is willing and able to pay for.  (And I haven't seen any indications that ".private" has been registered anywhere. Hopefully, it will be, but....  Another discussion from ServerFault.)

As for your venting, yes, Mac OS X Server isn't a "No IT" solution.  That written, Mac OS X Server is very easy to set up, in comparison with just about every other server I've ever dealt with.  Often vastly easier.  (But at its core here, you're now working with servers, which means you're expected to have a bit more knowledge about networking and security and related.  The servers are serving what makes it easier for the clients to be clients, after all.)

I do not prefer to use Apple Airport devices nor Time Capsule as network gateways for use with servers.  They make fine access points (APs), and they're good residential gateways, but they're not that capable as gateways for servers.  A server-grade gateway firewall can provide a VPN server end-point "upstream" of the NAT, among other features - and that VPN feature alone is often worth the price of admission, particularly given that VPNs and NAT are working at cross purposes.  And there are other benefits of having a good gateway firewall.

With the Apple WiFi device configured as an AP (what Apple calls Bridged Mode), DHCP services and the rest of the show are provided to the clients via the server, or via some other device that's connected to the network.   When configured as an AP, the WiFi device is basically transparent to the clients; the clients operate as if they were connected via wire, and with no intervening IP routers.

The location of your gateway firewall is key here.  That's the edge of your network, and what I use to distinguish inside or private from external or public.

On your gateway device...

The IP address and subnet mask on the external side of your gateway device are assigned by your ISP, either a static address assignment, or via a DHCP running at your ISP.

Your gateway's router address is received from your ISP.

Your gateway router might have a DHCP server pool.  If so, set this to a range of addresses in one of the private IP address blocks.  A subnet in 10.0.0.0/8, or in 172.16.0.0/12 or (less desirably) 192.168.0.0/16.

Inside your network...

Your internal hosts use your gateway router's internal address; your router address is your internal IP address of your gateway router.

Your Mac OS X Server is assigned a static IP address from your private IP address space.  I usually use low-numbered IP addresses for static hosts, as they're shorter and easier to remember.  The DHCP server gets a pool of addresses from, say, .100 to .254, if you're using the usual /24 subnet; that /24 is a different notation for the255.255.255.0 subnet mask.

Your client address is whatever you decide, static or dynamic, the router address is the internal address of your router, and the DNS server is the IP address of your Mac OS X Server; of your DNS server.

Workgroups are irrelevent to IP networking.

For a description of IP networking, see this IP Networking, Routers and Gateways article.

I'm working on a different presentation, but that's not going to be available on your schedule.

I don't have the cycles right now to write up and post a class in IP networking, particularly given what I have linked to (posted) is clearly inadequate for your understanding; that I've "missed" with what I've provided.

Please read the IP routing and DNS links that I've posted.  End to end.  If you have questions on those, please ask.  (That'll help you get some background, and it'll help me improve those articles.)

I don't know where that 172.16.0.0/12 address is coming from here.  I know some ISPs use weird IP address blocks (AT&T UVerse was using a 10.0.0.0/8 address space), but that 172.16.0.0/12 assignment is not one I've seen.  I'll guess that's something you've thought you needed to assign?  The external IP address is provided by your ISP.  That IP address is not assigned by you.  (You might have to enter it into the device, but it's an address value that you have acquired from the ISP directly from support, via an IP networking set-up FAQ for your ISP, or via the ISP's DHCP server.)

What you are calling "workgroup" is the subnet mask. What's usually called a "Workgroup" is something completely different. 

You have 255.255.255.0 listed as your subnet mask; that's what is (also) known as a /24 network, using the newer IP routing notation.  That means you have IP addresses .1 through .254 available to you, without either using an IP router, or without moving to a larger subnet.

Your DNS server address is the IP address of your DNS server.  If 10.1.1.1 is running your DNS, then 10.1.1.1 is your DNS server address.  (The sole exception here is the DNS server reference present on the DNS server network interface on the DNS server itself; that's 127.0.0.1; the IP itself; the IP localhost; the IP address of "me".)

If you don't have a secondary DNS server, then there is no secondary DNS server listed. 

If your IP router is at address 10.1.1.2, then the IP router address is 10.1.1.2 and not that 10.1.1.254. 

You'll need to enter the router and DNS server addresses in static-configured hosts, and you'll need to configure the router and DNS server addresses into your DHCP server for the hosts that get their addresses from DHCP.

And FWIW and without intending any insult, you can and probably should either do some reading on or watch some videos on, or attend some classes on IP networking, or you can offload this stuff and acquire more formal help.  When these networking configurations go wrong - and they do, I just ended up troubleshooting an error on one of my own networks after a power restoration this morning - you'll either get to troubleshoot the networking error yourself, or you'll want to call in help.  Having some background will help you troubleshoot this stuff, if that's your choice here.