What logs should I check for Alert Action troubleshooting?

You should have a log file called ActionExecutionalert.log in the directory, but this will by default populate when there are issues with the action.

Therefore, you need to adjust the logging level to get greater visibility, with the following KB explaining how

https://support.solarwinds.com/SuccessCenter/s/article/Adjust-logging-levels-with-the-Orion-Log-Adjuster

For the second part of your question, the email will be sent to a blank email address, as that is what the variable will populate - nothing.

For this, you could create a new alert, which looks for the Custom Property field to be empty and if so and has been for x hours, then use the Custom Property action to auto populate a default value in to this field. I suggest using the delay, so you can give the admin time to populate this data if they have added a new device.

You can also get a full log from the API via a SWQL Query

SELECT TOP 1000 [History].AlertObjects.EntityCaption AS [Triggering Entity]
     , [History].AlertObjects.EntityDetailsUrl AS [Triggering Deatils]
     , [History].EventType
     , [Actions].Description
     , [History].Message
     , [History].TimeStamp
FROM Orion.AlertHistory AS [History]
LEFT JOIN Orion.Actions AS [Actions]
     ON [History].ActionID = [Actions].ActionID
ORDER BY [History].TimeStamp DESC