I want to add some new staging sub-domains to my Sitefinity licence. My question is
1. While the request is being processed, will it affect the functioning of production & staging websites?
2. Will I need to deploy the new licence key file to the production site?
If you want to add staging sub-domains, you need to contact with Sitefinity team. Then, they will send you license file. Technically if you prod environment has valid licence, u don't need to upload licence file. You just need to apply licence to staging website.
Just for keeping everything consistent, i want to recommend to upload this licence file to all your environments.
P.S. When you replace the license file, you need to restart application pool of IIS