1.) Yes, you should be able to type in the IP address of the server directly and access it, even accross subnets. Though this is not always possible.
2.) Don't know.
3.) You can setup a local web server, and access it directly from the internal network via IP. However, setting up a domain name would likely not work easily. If you register the domain like a normal domain, then the server would have to be accessable via an external IP, which yours is not. You can just put an entry in the DNS that your school uses that would point to your server, but you would have to be able to change the DNS, which is sounds like you can't. You can put an entry into the hosts file that points to your server, but you will have to do that on every computer you want to use the domain with.
Rig1: System Specs.
Rig2: A8-5600K@4.4GHz / AsRock FM2A75 Pro4 / 8GB Corsair DDR3-1600 9-9-9-24 / HD7560D / Samsung DVD-Burner / 1.5TB WD Green + 3x3TB WD RED in RAID5
Rig3: Athlon X2 4200+ / M4A79 Deluxe / 4GB G.Skill Pi DDR2-800 4-4-4-12 / GT430 / Sony DVD-Burner / 500GB WD
Rig4: Phenom II x6 1605T @ 3.6GHz / Asus M5A99X Evo / 8GB PNY DDR3-1600 9-9-9 / GTX470 & GTX470 / Samsung DVD-Burner / 1.5TB Seagate