The catalog must be on a hard drive that is connected directly to the computer running Lightroom. The catalog CANNOT be on a network drive. If you want to work on the same images on both the desktop and the laptop then the easiest and most reliable way to work is to have the catalog and the images on an external hard drive that can be moved from one computer to the other. As far as mobile devices are concerned, that feature is only available if you subscribe to Lightroom CC. This is a subscription service that you pay monthly or annually. Payment of the subscription entitles you to use the latest versions of both Lightroom and Photoshop, and also entitles you to use Lightroom Mobile. Lightroom Mobile is a program for IOS and android mobile devices. The program works by creating collections of images on your computer and then choosing to synchronize those collections with Lightroom Mobile. Then you can edit images on the mobile device and those edits will affect the master images. But, again, this type of workflow is only available through the creative cloud subscription plan.
Put the catalog and the images on an external drive. That's the best way to work on a catalog between a desktop and a laptop.
To sync images to LR mobile on your phone you need to create collections, and sync the collections.