A few months ago my computer crashed and I lost all of my data. I haven't wanted to play GK again until now. Is there anywhere I could download save files? If it helps, I think I was on day 5 or 6.

The .cat doesn't seem to be downloading, when I click it it's just the box with the red X. Is this normal? How do I use it?

Try this:
Right click on it, choose 'save (target) as', save it to your desktop and then rename it to GKSG.cat

Okay, I'll need to be walked through this. I've done that and downloaded the files, what now? If I open the game will they just be there?

Ok, first, if you have any saves of your own at this time that you wish to backup, go to the game folder and move the GKSG.001, GKSG.002, etc, and the GKSG.cat file to some other place on your pc. Make a folder somewhere and cut/paste all of them to that folder.
Next, download all 3 zip files from Lafar's post #1, unzip them and place all the GKSG.001, GKSG.002, etc, (all 19 of them) in your game folder.
Lastly, place the GKSG.CAT file you downloaded (and renamed) from Lafar's other post also in the game folder.
And that's pretty much it, if all worked out you should see all 19 savegames inside your game

I put all the saves and the .cat into a new folder, and I opened the game, but they aren't in the game. What did I do wrong?

You have to place them all inside the game folder.

What is that? The only folder I found with the game was the install file in my GOG folder, but the shortcut on my desktop still works to launch it.

The game folder is the folder where the game was installed to, it should be
C:\Program Files\GOG.com\Gabriel Knight.
Just put the 19 savegames and the cat file inside that folder.
EDIT:
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If you are on Vista or Win 7 tick the 'Run as admin' on the game shortcut.
Right click the shortcut > choose properties > switch to the compatibility tab > tick the 'Run as admin' checkbox.
Without this is possible that the game goes looking for saves on the virtualstore folder and fails to see the ones on the game folder.
If you are on XP, this bit doesn't apply
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I put all the saves and the .cat into the game folder. Then I opened the game and clicked "restore", and there still aren't any saves. What should I change?
Also, the .cat says it's a jpeg image file. This is normal, right?.
No, I'm on XP.
I have to go now, but if you're still willing to help I'll be on here tomorrow.

I just tested Lafar's savegames with GK on my laptop - also XP - and they work.
Ok, so:
- Look at the attached pic to see you placed the 19 saves + GKSG.cat file on the correct folder.
- Make sure you renamed the cat file correctly to: GKSG.cat.
Edit:
No, it's not normal that it tells you that it's a jpeg file - that's likely the problem.
To make things easyer i've uploaded all the required files to mediafire.
Download them from here and extract them to the game folder, that will hopefuly get them to show up inside your game.
And sure, let me know if that worked. Otherise we'll try to figure out what's wrong tomorrow ;)

I should have realized sooner that uploading the files in a ready to use format was the easyer way, but yesterday was one of those very long days and it took me a while ;)
It's good to know you got the saves working.