No, you just need to connect the headset to the computer and adjust it to your liking using the Logitech software for best results. The 7.1 also works like this. (Via Bluetooth or mini USB)
Yes, you can plug the USB dongle into the PC and then connect the PS4 to the dongle via the jack. I also have this in a comparable setup. The dongle is connected to the PC and the laptop with the music is connected to the dongle via a jack.
No, this headset does not have Bluetooth, but connects via a wireless 2.4GHz USB dongle. Similar to the wireless keyboards and mice that use a USB dongle.