Hi,
As such, I'm looking for a little advice on debugging issues when apps don't work (yes, I've read what's on winehq.) I have the application KeePass (keepass.sourceforge.net) which installs just fine. When I go to run it, it just drops me right back to the prompt.
Pardon my ignorance, but why don't you use the Linux version instead? http://keepass.sourceforge.net/download.php lists a Linux port.
If you have the necessary tools to compile keepass, you can build the Win32 version from source and put a few debugging printfs into it, to see where it stops. This could shorten the debug time greatly. (Of course this doesn't work for apps where the source isn't available)