Your answer on number 2 explains why your router didn't work. You need to contact your schools IT and give them the router mac address. What happened is that your school is filtering all connected devices through the mac address and currently only your PC is given to them. Once you update your MAC address to your router, all internet traffic will go through that unit. From then on your computer and all other devices will connect to the router. If you ever want to update your router or remove it you'll need to contact your schools IT with the updated mac address of what ever device you'll be plugging into the ethernet jack.
Think of the mac address as a finger print. Every single device on this planet with a data connection either it be wireless or wired will have a differnet mac address and none will ever be the same.