They are probably allowed to...but it certainly isn't right. Good news though, they aren't allowed to use your laptop to spy on you when you're at home. One school did that, they got crucified.
However, I will say this, they certainly have a right to control traffic on their network, especially since you get access for free. But having you "register" your personal laptop, which I assume means install spying software on it, well, that seems unnecessary and quite honestly, intrusive. It's pretty simple to set up OpenDNS to block access to certain sites on your network, and you don't need special software to do it, other than the firmware that's already on their network routers. Their IT guy can certainly figure out how to do that. Schools and industries should stop using overly complex software, and realize there are often simpler ways of doing things. For instance, you don't need complex third-party software to stop people from saving things to a public computer, especially when Microsoft already offers software that does just that. It's called Windows Steady State, and (last I checked) it's free.
If they are saying they need to see your computer screen at all times, well, that's not okay in my book. It's your laptop, and if you want to play Solitaire, that's your business. If they are trying to restrict network access, then they can, but that might not be what they're doing.