Firstly, make sure you have ssh access to the server (dev, test and/or prod). Your ssh command looks something like this: ssh firstname.lastname@example.org or ssh email@example.com. Drush alias works by using this username and domain/ip address combination to login to the server and execute the command and then return back to your command line. In your project root, create a drush folder and inside it, create a sites folder. Now you put your site alias config in that folder.
Shutdown functions are functions that are executed after the current scope has been "shutdown". Say, you want to run a function after another function has completed execution, then this is the way to go. It is also a way to make PHP web requests somewhat asynchronous. In a Drupal context, say you are sending a request, that triggers a function. But you do not want to wait for the response or you want the function to start executing after the request has returned a response.