I think I mostly understand the symptoms you're describing now. I don't know why they are happening, though. :(
I don't use Geany very often, and my installation is pretty old, I think. But when I execute a Python script in Geany using F5, my cmd.exe window actually contains all the output, whether it's "normal" output (from print statements) or error output (exception tracebacks), so I get a chance to read everything before I press the key to close cmd.exe. I don't get a separate python.exe window at all.
When you have an error and your python.exe window goes away very quickly, that's just like what happens if you double-click the script in the Windows graphical environment.
I don't know how to fix your problem, but until someone more knowledgeable than I am can give you an answer, I at least can suggest a workaround: Open a command window manually (not from Geany), and run your Python script from there.
John Y.