Q: Need help putting in new library

Save the sound files to the desktop.

Highlight all contents in the music library in iTunes and click Delete. Confirm the removal.

Use home sharing to transfer or import the other computer's library.

Move the sound files from the Desktop back into iTunes.

Which operating system version? This is the first I have heard of system sounds being stored in an iTunes library.

Note that a library isn't just one thing. Typically a library (I prefer to say a collection since "library" is often used to refer to a specific file) consists of: 1 - library file(s) that store data such as what media files you have and where;  2 - media files which are the things actually played; 3 - associated support files such as artwork folders.  These are all stored in the iTunes folder a users's Home > Music folder.

If you want to get rid of a library, get rid of that folder.  If you want to replace it with a backed up library, put that library in its place.  Remember though that a library isn't just one file but a series of files and folders in a specific organization.  I don't know what is on your backup disc (or did you mean "disk" which is a hard drive, vs. "disc" which is a CD or DVD?) but if it is just media files then you don't have a library as such, just a bunch of media files.

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

iTunes 9: Understanding iTunes Media Organization - http://support.apple.com/kb/ht3847

Image of iTunes folder structure Dec. 2011 - https://discussions.apple.com/thread/3547807

see also:

Image of folder structure and explanation of different iTunes versions (turingtest2 post) - https://discussions.apple.com/message/13025536

It seems if I just copy it to the My Music folder, it doesn't go into a subfolder as it should but just stays single outside of any folder as a music file.

Copy it anywhere but the Music folder.  I think newer iTunes has an Add To iTunes Automatically folder but I am not familiar with its use.  Just drag the files to iTunes to add them.  iTunes should place a copy of the files in the correct folders unless you have this deactivated in iTunes Preferences (you shouldn't unless you really know how iTunes works and have reasons for not selecting it).  Highlight a file in iTunes then press command+R or right click > Reveal In Finder to check where the actual file iTunes is using is located.

If these are songs purchased from itunes you can add them to a second computer by downloading it again.

Downloading past purchases from the App Store, iBookstore, and iTunes Store - http://support.apple.com/kb/ht2519 - enabled with iTunes 10.3 and newer; not available in all countries; only apps, books (not audiobooks), music, and t.v. shows (not movies).  Discontinued items not available. For items not included in the iCloud list, or locations or computer systems where iCloud is not (yet?) available, you only get one download per fee paid.  Apple notes it is your responsibility to back up your purchases.

Select the store on the left side of iTunes.

Click on Purchased on the right side under Quick Links.

You can redownload your available previous purchases.

If they are not purchased from iTunes and you do not have iTunes Match then you will need to copy them to a flash drive or something and add them to the second computer manually.

Why do you want the check marks off?  If an item is not checked iTunes acts as though it does not exist for many purposes, but does keep it in the database.