Can Apple's Notes handle pictures pasted from Clipboard?
The answer to your question is yes, sort of. Notes in Mac OS can handle pictures either dragged in or pasted in from the clipboard. I didn't know this would work but just tried it and it worked perfectly.

My Notes syncs with both my iPad and iPhone through iCloud. In iOS the image pasted into a note on my Mac is represented by a paper clip and deleting that paper clip on an iOS device does not seem to delete the picture on Mac OS even after sync, so it doesn't work quite as well as I'd like with iCloud and iOS but works fine with Mac OS.

My guess is this capability is going to be expanded in a future version of notes and this is a place marker.