Some products require a key for evaluation and some don't. vCenter Server as well as ESXi run in eval mode without a key, but vCOPS requires a key.
You can use the purchased license on any installation you like, as long as you ensure each license is used only once. What you need to do to be completely compliant with the EULA - if you are going to use the licenses with your current setup - is to install the production environment without adding the licenses (i.e. run in eval mode), then migrate the VMs, remove the licenses from the pre-prod systems and add them to the production system. There's no need to re-issue the keys or anything else like this, the keys remain the same.
André