Windows Store apps (that's what they call Windows 8 apps officially) and Windows Phone 8 apps can't be synched as they are not the same. Some may work and look the same but you can't synch them.
Closing apps on Windows Phone is done by pressing the back button until you reach the start screen or other opened app. Once you do that, they are closed.
If you meant closing apps that have background tasks running like WhatsApp, Twitter, etc. You need to go into Settings - System and there is an option that lists all background apps. You can disable them there but you will need to manually enable each if you want them to be running in the background again.