Q: Computer Deauthorization/Authorization

Deauthorize all computers, then re-authorize the ones you use. You won't lose your music.

iTunes Store: About authorization and deauthorization - http://support.apple.com/kb/HT1420 - and another helpful post: https://discussions.apple.com/message/17828050

About deauthorizing specific computers - https://discussions.apple.com/thread/3809988 - Only, if you have physical access to the computer, otherwise you have to deauthorize all computers. You can only do this once a year.

If you're ever uncertain about any action with your computer, back it up first.  You should always have your music backed up.  Do not rely upon Apple to keep copies on their servers.  it was not until recently you could download a second time and there's no reason to believe that couldn't suddenly change again without warning.

There's different ways to back up so details will depend upon your overall preference.

Time Machine will do it automatically.

I use a cloning tool which lets me boot from my backup drive should my internal drive fail.

To do a proper backup you should do it to an external hard drive.  You can do a quasi-backup by making a second copy of files on your internal drive, but usually backups are needed when a drive fails (mechanically, computer stolen, fire, etc.)

Don't just back up your music.  Back up your whole hard drive, or at least all data files (media, including photos, documents, etc.).  My 100% clone means I can be up and running again a few minutes after drive issues and don't even need to reinstall software.

I really am not familiar with Windows computers, nor how to back them up, sorry.

As for backing up just iTunes, copy the whole iTunes folder to a hard drive (or flash drive if it will fit).

What are the iTunes library files? - http://support.apple.com/kb/HT1660

More on iTunes library files and what they do - http://en.wikipedia.org/wiki/ITunes#Media_management

What are all those iTunes files? - http://www.macworld.com/article/139974/2009/04/itunes_files.html