iCloud on my iPhone has Photos On, with iCloud Photo Library auto upload on, Download and Keep Originals, Upload to My Photo Stream on, iCloud Photo Sharing on.
You will not see the Live Photos for import in the Import panel, because My Photo Stream will already have transferred the still frame of the Live Photo. Do not turn on both "My Photo Stream" and "iCloud Photo Library". That will break the import of Live Photos.
With "Download and Keep Originals" enabled, you should be able to import the photos from your iPhone using USB, but first you have to remove the still versions of the Live Photos from your Mac's photo library.
And you need to solve the storage problem with your iCloud Photos Library. If your iPhone has been running out of storage, it may not have respected the "Download and Keep Originals" option and kept the video component of your Life Photos only in iCloud.
Is your Mac also syncing with iCloud Photo Library? Then you may have synced more images and videos to your iPhone than will first onto your iPhone. If you want to use "Download and Keep Originals", you must not have more photos and videos in the iCloud Photo Library on your Mac than will fit onto your iPhone.
Try the keep the iCloud Photo Library on your Mac real small, and use a second archive Photos Library on your Mac to save all your photos.
