Is there a way to, on my wireless network, have a domain assigned to a network page? For example, could I have "media.com" link to "192.168.0.101/sharedmedia.html"? It would also be good if I could "register" a network-wide TLD, such as "media.shared" (as with .onion URLs on TOR). I'm running apache 2 on a linux machine.
Sure. You need an internal DNS server. If your clients on your network use your internal DNS server (by say, assigning your internal DNS server to clients via their DHCP leases) you can create your own mini-world-wide-web. Create your own TLDs, overwrite other peoples TLDs, go crazy, create records for whatever you want.
You could watch the world burn by redirecting google.com to yahoo.com and google+ to facebook. The world would be your oyster, but only while in your office. As soon as you leave the office, everything would go back to being normal.