You photos will only sync from Lr Classic if you add them to collections and then activate the sync icon to the left of the collection name.
But you can completely pause sync if it is running.
Hover your mouse over the identity plate (top left) and click the triangle icon that appears to open the activity center.
Lightroom Classic CC always stores your files locally.
The sync with Lightroom mobile stores lower resolution versions of your image in the cloud, but as 99jon said, it's easy to turn this off.