Is it possible to work and process entire folders of images with LRC on the network - and

It's true that wherever the Catalog itself lives, must appear to the computer as a locally-mounted drive.

But the image files that are imported to that Catalog can live in a network location. I don't think a purely cloud location will work, but something like a server / NAS file-share accessed via the local network should be perfectly OK.

Photoshop offers ability to directly open from and save to Creative Cloud storage. But that is when accessing images one at a time. PS when used that way cannot access an entire image library in one go, as Lightroom Classic does. The extremely productive and integrated partnership of LrC front end + PS subcontracting certain tasks, fits within the practical restrictions of LrC.