Part of the problem is how a Mac, OS X, works. Once you close a window, the last window displayed for an app, clicking the Dock icon won't reopen a window for that app. With the app having focus, IE its name displayed in the blue bar (by clicking the dock icon), you click the name and select Open New Window (or something like that).
As for the sync with LR Mobile and the Web system that takes time. Syncing 656 images takes time to Upload and or check those previously uploaded images for any changes made to them in either the desktop LR or the mobile device LR mobile app and apply those changes, if any, to both LR DT, Mobile and the web interface. The time is dependent on your internet speed, mobile device connection speed.
It is not a problem with the internet speed. After all photos are successfully synchronized I want to quit Lightroom and then a ‘syncing 656 photos’ message appears at the top left of the Library module. This message will stay forever and Lightroom will not quit.
Could you please send me a LR Desktop diagnostig log - best as a private message with a downloadable dropbox link.
You can trigger the log via LR Desktop preferences -> Lightroom Mobile and when you hold down the alt key you will notice a generate diagnostic log button