Easier formulated: My question is, do I not have to worry developing and deploying asp.net core apps locally about license fees and is asp.net core safe to run locally on production machines.
I do not understand your question or concern. Windows Server requires a license to run. If you build an application that targets Windows Server, then anyone that installs/hosts your application on a Windows server needs a license to run Windows.
GNU is a license for sharing software with rules related to modification. You can offer your source code under a GNU license if you like.