Why wouldn't a dorm room wireless router set up?

Strictly speaking, one wireless network should work just fine. Or

2-3-10. But once you get enough networks all sharing the same spectrum, no one's network will work, including the dorm's own wireless network.

Does your laptop show a dorm wireless network? If it does, try it.

Define sucks. If you're expecting to run Bitorrent, massive cloud server applications, or movie downloads, yeah it probably sucks. For email and web browsing, it's probably just fine. See the dorm network terms of service for the fine print.

If the skool ResNet registers the MAC addresses of all connected devices, it won't work unless you register the router. The problem is that the skool needs to know about all the devices that connect to its network. Wired and wireless routers hide the MAC addresses of the connected computers, so the skool sniffers can't see them. So, in self defense, the skool probably refuses to allow routers.

However, you might be able to set it up as an access point which does NOT hide the client computers MAC addresses:

Get a router that allows you to spoof (change) the WAN MAC address.

Then register the PC like the school wants. Once the wired connection is working directly to the PC change the WAN MAC address of the router to match the PC's and you should be good to go.

How the school may block the above method is if they are using a web based checks and balance system. Much like many public library systems where you must read a web page and click on an acknowledgment icon that you will follow the school rules at the start of every session or say every 24 hours. If they use that kind of verification then no a router will not work but as Jeff mentioned a dumb wireless access point will work as the text acknowledgment system will pass through the access point and work just as if the PC was wired.

That won't work. On the school network, the monitor/sniffer software compares the MAC address with the user login. If the login is used with a new MAC address, it is logged, but usually nothing happens. However, if a MAC address appears, and there's no associated login, questions might be asked, especially if it's the source of excessive traffic. In the dorms, ResNet is not so picky about security and usually does not have access to the schools authentication system.

I'm not going to get involved in supplying workaround and bypass methods.

Note that not every skool dorm bans routers: Some allow only access points:

Locally, UCSC does not exactly ban routers, and then only in specific dorms. Where allowed, they ask that they be setup by someone with a clue:

here's how UW/Madison addresses some of the issues - registering a router, xbox, game console, etc -

"Wireless routers, fresh out of the box, are configured to act as a DHCP Server and issue out DHCP addresses. While this works fine at home, it is not compliant with our network and will cause major network disruptions/errors for both the router owner and all students who live in that same area of campus. If you choose to configure your router yourself, and it isn't configured correctly, we will take necessary actions to shut off the port until your router is brought in to our office and is correctly configured."

Does this make sense?

I think they're protecting themselves from the people who accidentally plug a router's LAN port into the school's network, thus exposing the router's DHCP server to the school network. Major disruption is possible and likely.

If the router's WAN port is connected to the school's network, there is no DHCP exposure, but then other issues/restrictions/policies may come into play.

I think they mean "connections and DHCP addresses".

Yes.

Anyone connecting to a rogue router at a college would appear to be in dorm room X, according to the monitored traffic, perhaps exposing the router owner to difficulties because of unauthorized traffic.

Depending upon what was being accessed, certain servers' inability to initiate traffic back to that laptop because it was behind a PAT router would be problematic and hard to discover.

This is something that a competent enterprise grade switch simply won't allow, everything but the low end SOHO grade switches have the capability to only deliver DHCP packets to specific ports, and therefore a disruptive port can't disrupt anything but itself.