1. NSA does not "hack" computers, they intercept information. Unless, of course, you gave them reason to investigate you.
2. If you are sure your computer was accessed externally (I'd like to know how you came to know that), it's likely that your brother was just messing up with you remotely, through some remote connection he authorized in your machine (again, not hacking).
3. Not having permission to format the system drive should be pretty obvious, since the system is running on it. You cannot expect the system to format itself into oblivion. You should format it from outside windows. No hacking trick can prevent you from formatting a hard drive.
4. If your computer had an OEM windows version installed, you may be able to get a backup disk from the manufacturer. If it had a full retail license, you may download a clean windows image from torrents and the like, burn it to a DVD and install with your old license. That would not be illegal. You may even request an install disk from Microsoft, if you have a valid license.
5. If you only messed up the registry, a Windows 7 boot disk may indeed fix it, though a system running with a tampered registry is hardly reliable. Reinstalling windows would be the best solution.
6. Once you get everything running, I would advise you to install a good firewall, so you know can manage all external connections.
Still, you will never be able to prevent NSA from spying on you, since they can intercept your external communication. You should be able to prevent local access, though.