I downloaded the 2 install files, one exe and one bin file, as well as the two patches.
The exe will launch and install, but then when I hit launch, I just get a generic GrimFandango.exe has stopped working error.

I'm assuming it has something to do with the BIN file not doing what it's supposed to? It seems like it isn't being used at all. I'm a bit confused about how I'm supposed to install.

Here's the problem details:

Problem Event Name: APPCRASH
Application Name: GrimFandango.exe
Application Version: 0.0.0.0
Application Timestamp: 54c6d0ec
Fault Module Name: atioglxx.dll
Fault Module Version: 6.14.10.9236
Fault Module Timestamp: 4b2aa66c
Exception Code: c0000005
Exception Offset: 0000b212
OS Version: 6.1.7601.2.1.0.768.3
Locale ID: 1033
Additional Information 1: 0a9e
Additional Information 2: 0a9e372d3b4ad19135b953a78882e789
Additional Information 3: 0a9e
Additional Information 4: 0a9e372d3b4ad19135b953a78882e789

As an update, I downloaded all of the installer files again, but is giving me the same error.
This time around it asked me to link to the BIN file, which I did, but it didn't seem to change anything.

You could try downloading and installing the game with the GOG Download Manager, that does everything automatically... as long as you set it to install the game when the download is complete.

It sounds like the game installed ok, but then the game crashes because of some issue with the ATI (AMD) drivers (as the Fault Module Name line suggests).. With the OpenGL module specifically.
You could try to manually download the latest drivers from AMD's site and update them for your PC.
Also choose to restore all factory defaults for your graphics card from the catalyst control center app, or at least revert to defaults for the OpenGL settings (eg don't use triple buffering etc).