A: WiFi routers can be scanned for by their wireless signal (the electromagnetic frequency that is easily triangulated), and can by way of device type on the network.
Improperly configured, such a router could easily be a potential security vulnerability to the network and the devices connected to it. Even if properly configured, it would likely interfere with the optimal performance of the network as well.
Keep in mind that when using a school or business network, it is their network, not yours. They get to make the rules. Lastly, should the router in any way interfere with the network or devices on it, or access any network data, that could be seen as a felony by the very gumpy folks at the Department of Homeland Security.