iCloud Photo

Yes, if both devices have iCloud Photos turned on.

iCloud Photos is a syncing service. Any photo you delete from a device where iCloud Photos is turned on will also be deleted from iCloud and from any other device connected to iCloud Photos. You can potentially reduce the storage space required on a device by turning on Optimize Storage. See: Set up and use iCloud Photos - Apple Support

Apple does not currently offer cloud-only storage for photos. If that is what you want or need, you may want to explore 3rd party options such as Google Photos or Dropbox.

If you want to turn off iCloud Photos on a device, see the information in: Get help with iCloud Photos - Apple Support

Yes, if you turn off iCloud Photos on your desktop and leave it off. If you turn it on again, the photos on your phone will sync with your desktop.

If you want to turn off iCloud Photos on a device, see the information in: Get help with iCloud Photos - Apple Support