Really? When I try to just rename the .dll, I can't run my app
It's possible to rename the dll/dylib and use it with your app, but I didn't say it would be easy.
On Windows you have to link directly to the *.dll file (which I can't remember how to do off the top of my head, but I'm sure Google can help you out). That is, don't use the #pragma(comment, lib, "TurboActivate.lib").
On Mac OS X you might have to use some developer tools to change the "internal" name of the dylib so the correct file is loaded. You'll also have to changed the embedded rpath.
In short, it's probably more hassle than it's worth.
One more question - do the dynamic libs have any other dependencies? I.e., on Windows, does TurboActivate.dll require the MSVCRT dlls? I just want to make sure I can ship my program without the user needing to install anything else.
There are no external dependencies. See Requirements for TurboActivate