Pete,
The password is stored in your keychain - it sounds like there's some issue with it or the security of your user in general (the way it works is the keychain trusts WeatherCat because it is signed with an Apple certificate so you don't have to authorise WeatherCat to access it).
Check the state of your keychain with the Keychain utility (Applications->Utilities->Keychain) and that WeatherCat has access.
The second issue might suggest either you are out of disc space or a permissions issue (WeatherCat has no permission to write to disk) - again possibly an issue with the user you are logged in as.
First thing to do, is quit WeatherCat then delete your existing WeatherCat folder (maybe it was downloaded/installed as another user), download it again and run it (as the user you are logged in as).
If that's still the same, you can create a new user, log in as that user, download WeatherCat and run it as that user - that'll tell you immediately if it's some problem with the user account you are using at the moment.