Build master in release
Use Webhook with CodeBuild to build any branch:
https://docs.aws.amazon.com/codebuild/latest/userguide/sample-github-pull-request.html
Get nupkg file from bin/release folder and publish it to nuget.org
CodeBuild can run any arbitrary command in buildspec:
https://docs.aws.amazon.com/codebuild/latest/userguide/build-spec-ref.html#build-spec-ref-example
Add that same file as a release to the github repo
I am not sure, you may need to use a third party binary to achieve this which is run in buildspec
start a custom *.exe
Will you run this during build or deploy? If build, you can again run the command in buildspec, if deploy, you need to use a deploy service like CodeDeploy.