If you want the css to be propogated to all the files, you will need to apply the template in Dreamweaver and then upload all the modified pages. If you make changes to a template and then upload just the template, the changes will not be applied to the pages that are based on the template.
I would recommend making a backup copy of the database and the site files before making any changes.
You can either download all the files and apply the changes, and upload the files and see how they look by testing there or as you said, download all the files, and import the database and change the connection file and test on a local machine.
PowerStore is meant for deploying on a server and getting it all installed there with minimal hassle, so that is why there is no directions for copying between two different servers.