I have no idea why the old photos are showing up. As Yul Brynner said in
The King and I "'tis a puzzlement!". But to explore that issue further...
- By any chance have those deleted images ever been on your desktop?
- On your Laptop in Photos under Library > Recently Deleted look and see if those images appear there.
- Look in the same place on your Desktop.
- On your Desktop see if those images are listed under Imports.
In answer to your other questions:
- Assuming all of your devices have the correct settings they will automatically and transparently synchronize. Think of it this way: the MacOS and iOS devices are all mirror images of the iCloud Photos Library. The settings I recommend for MacOS and iOS can be seen in these screenshots.
- There is no reason to open the iCloud Photos Library unless you are on a strange computer (ie. not linked to your Apple ID) and want to show someone your photos.
- Yes, you could replace the Desktop's Photos Library with a copy of the Laptop's Photos Library and that should work.
