This is my first time buying a game on gog, and I am wondering what I have to do to get this game running. I downloaded the files onto my desktop and placed them all into a folder, and when I try and use the first download it has an error message.

"This file could not be located in the installer's folder. Please make sure that the file is in the correct folder, or specify its current location."

So what did I do wrong?

Not sure what went wrong, but try this - download and install GOG.com Downloader http://www.gog.com/downloader
Then turn GOG.com Downloader mode ON (see pic), click on Windows installer to download the game.
Go to folder with downloaded game and click on setup.exe.

The downloader freezes every time I try and use it. I got about thirty minutes through the download when it froze the last time I tried it. This is what it asks for.

Try different browser to download parts 5 & 6.

The last download (setup_jade_empire_2.0.0.4-5.bin) isn't finished, hence the installer didn't find the file. Redownload "Windows installer, English (part 6 of 6)", save it to your desktop and run the installer again.
Make sure your files are correctly named:
setup_jade_empire_2.0.0.4.exe
setup_jade_empire_2.0.0.4-1.bin
setup_jade_empire_2.0.0.4-2.bin
setup_jade_empire_2.0.0.4-3.bin
setup_jade_empire_2.0.0.4-4.bin
setup_jade_empire_2.0.0.4-5.bin

Any difference will stop the installer from finding the .BIN files.
If you don't see the extensions (.exe and .bin) they're hidden by Windows. Here's how to make them appear:
http://windows.microsoft.com/en-us/windows/show-hide-file-name-extensions

Yes, it seems the final download hasn't finished properly, for some reason. Some Googling suggests that the .crdownload or .crdownload.file extension is used by Chrome for files that are in the process of downloading.

The file seems to be about the right size, so you could try removing the .crdownload extension and trying the installer again (it has a built-in file integrity check), but it may be simplest just to repeat the download.