You asked a few questions, and therefore I answered those that I can base on the information provided.
Crossrider is a cloud service, and it requires Internet access to download code and extension resources.
In general, once the extension is installed and resources are loaded, the extension runs locally and periodically visits the Crossrider site for updates. Consequently, Crossrider (CDN) server dependency occurs primarily during installation and periodically thereafter, because extensions are performed locally after installation. This model comes with several benefits, including automatic code / frame updates and a statistics panel. Of course, if this does not meet your requirements, there are other stand-alone platforms that you can use to create the extension.
Regarding resource loading, this usually works without problems. I am happy to take a look at your script, but for this you need to specify the identifier and versions of the browser and OS that you use, and I will investigate.
[ Disclosure : I am an employee of Crossrider]