I currently run an older version of Lightroom on my Mac. I have thousands of images organized in a library and file system that works well. If I transition to CC and Classic, will they (or will Classic) recognize my existing organization and presets?
Lightroom Classic is the continuation of the Lightroom you have been using, so it will recognize everything. It's the same thing.
"Cloud-based Lightroom" is called "Lighrtroom" by Adobe (forget CC) and is a new application which is like having Lightroom Mobile on your Mac/PC. You can "migrate" a catalogue into it, and it will copy your files up into its cloud storage. From that point on, it ignores your file system - its originals are those in the cloud and it doesn't have folders like your filesystem. But it will recognize collections which were in your catalogue, and keywords too (though it flattens any hierarchy). You can import your presets. It's a very different program.
I went through this transition to Lightroom (Cloud) a few years ago and the process was pretty straightforward. The app has a feature to import an existing Lightroom Classic Catalog. Takes a while to get the images to the cloud and retains collections. You upload presets separately, IIRC.
Once your images and presets are in the cloud, though, they are available full-quality from any Lightroom client (Mac, PC, iPad, phone). Some features such as printing and tethering still require Classic, but I've been happy overall with the transition as I prefer the simpler interface in Lightroom (Cloud), love having my images always available from every client, and the develop functionality is really close to parity with Classic at this point.