I have only used the DS4 on an SP2, but it should perform similarly on an SP3
Yikes! 10cm! The range is definitely further than that. I have used it from my couch with the SP2 connected to the TV (about 3m). I believe that DS4's advertised range is 10m.
You can connect the DS4 to a PC using either BT or USB. I have not used it with a USB connection myself, but it should not be a problem.
I have not had any issues with the games I have used (mostly Steam). I have not used it with any metro apps.
You can, but I really don't like it. It tracks fine, but trying to click causes the cursor to jump resulting in not actually clicking where you want. Also, the gestures are not great (e.g. two finger scrolling). This could improve with new versions of DS4Windows.
I am not sure of the battery life and It varies with vibration use, but it doesn't seem to be the greatest. It only has an 800mah lithium battery. It's not horrible, just not great. There's a "soft" on/connect button between the sticks. The DS4 turns off when disconnected from the computer plus I have set a 10min disconnect. Also, it does not come with a micro USB cable (used for charging).
Hope this helps