Is there a way to download older versions of blender without replacing the current one I have installed? I can't get the grease cut tool to work. I attempted this course a while ago on version 2.76 or 2.77 and it worked fine. So I'd like to download that so I can take another stab at this course, but still keep the 2.79 version that I have.
Yes you can definitely do that - I do frequently. In fact I wrote a blog article about it not too long ago. Just download any version of Blender from here as a zip archive. Unzip it and double click the Blender executable. You can have as many versions of Blender as you want if you use the zip archive "installation" method.