Moving imported photos

If synchronization is enabled then when images are taken and added to Lightroom (for the cloud) they should automatically be added to your computer. At least that is the way it works on my computer.

If you move the photo within Lightroom iteslf by dragging the image and dropping it onto the target folder in the Folders Panel, then nothing you need to do as Lightroom will update its catalog with the new location. However, if you move the image using the file system, Lightroom will not know where you've moved it so the image will be reported by Lightroom as "missing". You can subsequently relink Lightroom to the image in the new location, but it's a lot easier to do the move using Lightroom itself.

I don't understand the second question. Is the image in question one you've synced from Classic to the phone (via the Adobe cloud)? If so, I assume you mean what happens if you delete it from the LrMobile app on the phone.  The same applies if you've synced the image from the phone to the cloud, and thus down into Classic. If so, the image is removed from the cloud, but remains in Classic.If you mean something else, please clarify.