Server Set-Up w/ Airport Extreme, Static IP Assistance

For examaple, for the DNS piece, do I need a registered domain name

'need'? No. Want? prefer? desire? sure.

will something like www.mycompany.com work?

If you mean literally 'mycompany.com', then mabye, but wouldn't you prefer your actual company name?

If I have one, should I use it?

Well, what are you going to do with your own domain if you're not going to use it?

Do I need to modify the nameservers within my web-hosting control panel (if I use a large 3rd party web hosting solution).

No. Your third party web host has nothing to do with this (unless they manage your public DNS, too)

see a lot of "example.com"s thrown around...should I just use that?

No. 'example.com' is just that - an example. You're expected to replace this with your own domain name.

I want to be able to access our server remotely, so I believe, DNS is needed.

DNS is needed if you want to given humans an easy-to-remember way to get to your server. There are only a few services that need DNS, but you haven't said what services you want/expect to run to know whether that's needed.

1) Airport Extreme settings config (incorporating my new IP, subnet mask, gateway and 2 DNS servers I was provided with today by my ISP). Where do I insert these pieces?

In the WAN settings of your base station (assuming that your base station is at your network edge (i.e. where your ISP link plugs in).

2) Server admin settings for adding the new server (I removed our previous one from the list in server admin), as we were only using it for File Sharing and those same files still reside on the hard drive and will be easy to make "shared." If I click the "+" to add a server, what "Address" do I provide? etc etc etc

Server Admin is designed to administer multiple servers, so you'd typically add each server that you want to manage. If you only have one server, that's what you add.
Additionally, you should be running this from within the LAN, so you should enter the LAN address of your server. If you set the public (WAN) address and it works, then something is seriously wrong with your server (i.e. anyone could hit your server and administer it).

I know this is quite a large ask and I respect the time of all the community members here! But I'm hoping there exists a guide somewhere or someone can break it down very simply for me.

There isn't one guide to rule them all because there are many, many ways to setup your server and your network. What does exist is a number of guides for individual components, so you need to consider what you plan to do with your server (e.g. directory server? LAN file server? public web server? etc.) and follow the guide for each.

Apple's Server Documentation is a place to start, but it will cover more than you need at this point.

we'd like to use our server mainly for file sharing both inside and outside of the office

That's still a rather vague description.

For example, do you care about security? Do you care if random internet users hit your file server? Do you care if the files are intercepted en route to the remote user?

Maybe you don't - if you're only hosting public data anyway then that's quite possible, but if you're storing company-confidential data then you do NOT want to run a public file server. You WILL be probed, you probably WILL be compromised and your data probably WILL end up in some unknown user's hands.

If all the data is for internal company use only but you have remote users that want/need the data, then what you want is a VPN that allows users outside of your network to connect to the VPN server and then securely access corporate resources. In one step this will take care of your wiki, file server, mail, calendar, and everything else.

To that end what you should do is setup the server on the private LAN, ignoring any concept of 'public' vs. 'private' content. This will involve setting up your own DNS server for your LAN and configuring each of the services (web, mail, ical, etc.).

Once you've done that you can enable the VPN service, configure your router/base station to permit incoming VPN connections and you're done.

Our company website, www.wearesewcreative.com is hosted by MediaTemple, and I would like to use this for DNS settings.

I don't know what you mean by 'like to use this for DNS settings'. Do you mean you want to continue to use MediaTemple for your web site? That's fine. Nothing I've suggested so far makes any difference to your public web service.

How do I establish the server on this network and allow it to be accessible in the office and outside of the office?

Step 1: set up the server
Step 2: enable the VPN service
Step 3: configure the base station/router to permit incoming VPN traffic and route it to your server.
Step 4: there is no step 4 (but don't be blinded by the amount of work in step 1, above)

When I enter server admin it asks for "Address" of the server. What address to I add?

Server Admin is trying to talk to the server to configure it. It's highly unlikely (or desirable) that your server should be administered via the public internet (you don't want any old hacker reconfiguring your server from afar), so you should only EVER setup Server Admin on the machine's private (LAN) address.

Do I need to give the server it's own address coming from the Airport Extreme?

Your server needs its own static address within the LAN. For a typical LAN you might have your router address at 192.168.1.1 so you could set your server at 192.168.1.2 and use range such as 192.168.1.100-120 for DHCP clients. That still leaves plenty of room for other devices such as printers that you might put on your network). If you follow that scheme then 192.168.1.2 (the LAN address of your server) is what you'd put into Server Admin.

If it do this, will the server be accessible outside of our network?

Define 'accessible' please. If you mean can you run Server Admin from afar then the answer is absolutely not. You don't want this. Ever.
If, by 'accessible', you mean individual services such as web service, ical, mail, etc., then no, not automatically - you'll need to configure your router/base station to allow specific incoming traffic (e.g. port 80 for web, ports 25/110/143 for mail, etc.) to route to your server.
For remote access to your Server Admin you want to use a VPN.