Is there any program that I can use to open my .STLs with and interact with them to simulate how each turn would look?

You can look at STL files using MeshLab or Netfabb Basic. Both are free - just Google the names.

I highly doubt that any program like that exists. Even a very advanced CAD package like SolidWorks doesn't really have an easy way to simulate the turning of a puzzle, even before the file type is converted to STL files. In SolidWorks, you have to either insert a Body Move/Copy feature for each turn you want to make, or insert all the pieces into an assembly, and change the mates each time you want to make a turn.

Really, the best way to see how a puzzle turns is to get it printed!

-Eitan

I have exported STL's to blender, and then used that to move things around, but it's still a pain...

Indeed, but $60+ bucks is an awful lot to invest up front, especially as a college student (as you can understand).

Sometimes the real world is best.