Welcome to the Ad Tech world! The short answer to your questions is yes.
As long as all of the specified sites are approved within our system, any tag within your account should function across all domains. Whether or not you decide to implement a single tag across multiple sites depends on how much control you want over the performance and reporting of each tag/site.
The most simple implementation would be to run one tag across your entire network of sites. This allows you to easily change the target CPM’s for your entire network by adjusting the floor on the single tag you are using. This implementation will require the least amount of time.
However, if you wish to have more granularity into your reporting breakdown as well as your ability to target different CPM’s across individual domains, I would recommend creating and labeling new tags for each. This can be helpful as the demand we are integrated with will likely value the inventory from each site differently, and having separate tags for each zone will allow you to optimize your overall performance better, increasing your overall yield.