This issue has been solved. Posting back here for public education.
The main problem associated with this is that no connection info was set up in SmartConnector.WeatherExtension.Test's App.config file.
When you run the test, it will try to connect to Smart Connector and set up an EWS server; however, it cannot do that if no connection info was set up. You have to go to the install directory for Smart Connector (will probably be C:/Program Files (x86)/Schneider Electric/Smart Connector) and open "Mongoose.Service.exe.config" in a text editor. Locate the <connectionStrings> element and copy the entire contents between the opening XML tag and the closing one </connectionStrings>. Paste that into the App.Config file, and do the same for the encryption key setting (included in the <appsettings> element). Refer to the attached picture.

However, if freshly downloaded from the repo, you will run into more issues after this one because a couple things need to be set up that are not. The next error will have to do with licensing, most likely. You want to disable licensing, so that a valid license file will not be required to run the processor. Simply have this region of code uncommented in the WeatherProcessorBase class:
#region IsLicensed - Override
public override bool IsLicensed => false;
#endregion
After all this, one will run the ExecuteTest (for SetupProcessor) with seemingly no unit test errors, and an EWS server will be created in the SmartConnector portal. However, the server will be stopped and cannot start. Looking into the logs will show an error that "SmartConnector.WeatherExtension.dll does not exist". Because of the custom EWS implementation, a DLL for this extension must be present in the Smart Connector Install directory. Follow these final steps and everything should run without a hitch:
1. Build the "SmartConnector.WeatherExtension" project after you have removed the need for licensing
2. Open Task Manager and Stop SmartConnectorService
3. The DLL produced by your build can be found in SmartConnectorSamples-master\SmartConnector.WeatherExtension\bin\Debug. Copy that dll into the SmartConnector install folder
4. Start the service back up
5. Run your unit tests