Can i use Lrc on my pc and mobile, one at a time?

The mobile or web apps can see and edit only the photos stored in the Lightroom cloud. So if you use Lightroom Classic and want to use the mobile/web Lightroom apps to edit photos from a Lightroom Classic catalog, you have to manually enable cloud syncing for those images in the catalog, or for a collection containing those images.

Lightroom Classic does not upload full resolution originals to the cloud, only Smart Previews.

But, when you import images into a mobile Lightroom app, or capture photo using the camera in a Lightroom app, Lightroom Classic can download those full resolution originals to your computer and import them into the catalog in which you enabled cloud sync.

Thanks 

LrC does not run on mobile. The mobile app, named Lightroom (or Lr) should run on your mobile device at the same time that you are using LrC on your PC. Why don’t you try it?