I would also have to have a monthly subscription to CC? Am I understanding this correctly
Yes you are correct.
If you own Lightroom 6 you cannot sync your images from your Desktop to Mobile(after the 30 days trial expires) as it requires Creative Cloud subscription version.
If I own LR 6 (disk), can I not also have mobile?
You can now use Lightroom Mobile on your Phone/tablet without having the CC subscription as it is now available as a standalone image editing App.See Download Lightroom for mobile | Adobe Photoshop Lightroom for mobile