Lightroom sync

There is a setting in the Lightroom app you need to turn off. It's in gear icon (either top right when you hit the left arrow at the top or in the ellipsis menu when browsing images) -> import. Turn off all the "Auto add from camera roll" options. If this is turned on, it will automatically start uploading all your images on your devices Photo roll, especially if you use iCloud Photo Library. It is very annoying as you have to do this on every device you own otherwise you can even end up with double or triple uploading. This feature is badly implemented unfortunately and interacts badly with iCloud Photo Library being turned on. You'll have to manually delete all images already uploaded or already imported and waiting to upload.

To make sure I'm understanding you correctly, are you saying that when you open the LrMobile app on your iPhone all the photos that you already have stored in your Adobe Cloud are downloaded to the iPhone? If so, then that is entirely expected...the whole point of the Adobe Cloud is that all your photos are automatically made available on all your connected devices. But note that does NOT mean that the photos are downloaded in their full original format and size, they are only initially downloaded as quite small previews, which will not take up much space. When you try to edit a photo then LrMobile will need to download either a copy of the original, or a Smart Preview (there's an option on the LrMobile Settings>Cloud Storage & Sync tab which allows to to "Only download smart previews", which are much smaller than the original file but are usually good enough for light editing). With that option enabled, the only time you'll have an original downloaded from the cloud is when/if you want to do an export larger than the 2560 pixels of the Smart Preview).