Changing the project name does not change the file name; but you can rename the project file at any time in your file manager (the only side effect of this is that Tropy will not open your project automatically, because it will look for the old file name, so you just have to open it manually after you rename or move the file).
Starting from scratch might help, but the error message looks like a deeper issue. We are not supporting Windows 7 'officially' (i.e., we are not running any tests on Windows 7) so this could be a fundamental problem.
For testing, I would suggest to remove Tropy and re-install it, as you suggested. Then create a new project at the default location and import only a handful of photos (ideally from your local drive). As long as you don't move your real project or the photos you can still open it so you're not losing any data, this would mainly be trying to see if a 'minimal' project/configuration works fine on your computer.
Also, please note that in order to use the project from two different computer, you need to make sure that the external drive is assigned the same drive letter on both computer.