What do we specify as the address of the source in the NSDI Spatial Data Source Register and in the NSDI Metadata Catalog if our spatial data or more detailed information about them is not published on our institution's website, but on another institution's website?

According to the NSDI Metadata Specification v.3.1, the address of the source must be specified if it exists, that is, if a URL is available that provides more information about the source and/or the service related to access to the source. In case your spatial data is published on e.g. on the county page, it is listed as the address of the source, and it is not necessary to publish them on your institution's website.