This all is expected behavior since there is no real workflow to work across both versions. However, there are some things you can do and try.
With LR CC the original is (always!) synced to the cloud and there is no smart preview to sync down to LR Classic.
If you import an image to LR Classic CC first and add it to a synced collection, a smart preview is synced to the cloud. If you now import the very same image again using LR CC - maybe because you'd like to have the original in the cloud - the 2 images should be linked together and the original raw should not be downloaded to LR Classic.
How do I work with previews and a common catalog across LR Classic and LR CC, or do I need to keep a local catalog for Classic and only synchronize a single small collection between the two?
Can I keep all my originals in the cloud and still use LR Classic or does Classic force me to have everything locally?
The only way you could "work with previews and a common catalog across Classic and LRCC" is by doing all the importing/syncing in Classic (as that will only upload smart previews, not originals). But doing it that way means your originals are stored locally in Classic, and you never have any originals in the cloud whenever you want to export an original from any of the LRCC apps/devices (e.g. phone or tablet). You also can't "keep a local catalog for Classic and only sync a single small collection between the two"....as soon as you enable sync in a Classic catalog it will immediately download every original in the cloud, as you have already discovered. Whilst you can selectively upload images (as Smart Previews) from Classic to the cloud, there is no way to tell Classic that you only want to download selected images or collections. It'll download everything that doesn't already exist in it's catalog.
So the answer to your last question is "yes" and "yes".....yes you can keep all your originals in the cloud and still use Classic, and yes working that way would mean that Classic will download everything locally (and well as retaining everything in the cloud). That is in fact how I operate....everything stored locally in my Classic catalog, but all files are synced to the cloud (as originals, not as Smart Previews).