I procured a support subscription to 1 Rancher Management Server and a 20-node starter pack for my Production environment. Does my support subscription cover issues in my Dev/Test and Staging environments as well?

No. In examples such as this one, customer is required to procure license(s) for any additional Rancher Management Server installations for which support has been sought. In this example, whilst Rancher Support will help customer only on a best effort basis, for their issues in Dev/Test and Staging, they will recommend a conversation for the customer with their Rancher Account Executive for procuring additional licenses at the earliest.