Will an Xbox One S controller work via bluetooth, and via the Play & Charge kit on Windows 10?

Yes it will. You can connect the controller to your PC via WiFi or Bluetooth and can connect via your plug and charge kit to charge the batteries while you're playing.

But for playing on PC you don't even need the play and charge kit, you can just connect your controller via micro USB cable to your PC even if your batteries are drained as while connected via USB it wont use batteries.

Hi,

In short, you can't. Unlike Xbox One or Series X|S controllers, the 360 wireless controllers are always wireless. The Play & Charge kit simply charges the battery, it does not send data.

The only way to use an 360 wireless controller on a PC is via an Xbox 360 gaming receiver for Windows.

You can use a regular Xbox One controller on an Xbox One S console, and you can use an Xbox One S controller on a regular Xbox One console.

All controllers connect to the Xbox console the same way, via proprietary RF.

If you are confused because of the addition of Bluetooth in the Xbox One S, just know that Bluetooth is only built into the Xbox One S controller (there is actually no bluetooth functionality in the Xbox One S console itself). The Bluetooth in the controller
is just a secondary method of wireless connectivity designed solely for connecting the controller to a Windows 10 PC via Bluetooth (without the need of a
USB adapter). It connects to the Xbox One/One S the same way as the original controllers do.

As schnurren already mentioned, the Play & Charge kit only provides power to the controller. No actual input data is transmitted via the Play & Charge cable.

You will need the
Microsoft Wireless Gaming Receiver, and Windows 10 will automatically download the necessary drivers from Windows Update.

DO NOT confuse the Microsoft Wireless Gaming Reciever with the newer Xbox One Wireless Gaming Receiver for Windows 10!! The black USB plug receiver that looks like a big USB flash drive won't work with the Xbox 360 controller. You will need the slightly
oval shaped Wireless Gaming Receiver that connects to an USB port via cable.

Alternatively, instead of
buying that Wireless Gaming Receiver, you could as well
buy a Xbox One controller from that money, and connect that controller with a standard Micro USB cable or via Bluetooth (if supported by your PC).

There is no Bluetooth radio component in the Xbox One S console itself.

The controllers connect to the Xbox One S the same way the controllers connect to the original Xbox One: proprietary RF. You could technically connect an Xbox One S controller to an original Xbox One, and an original Xbox controller to the Xbox One S.

The Bluetooth functionality in the new Xbox One S controller is specifically made for connecting the controller to a Windows 10 PC with the "Anniversary" update, which has the drivers needed for the controller when connected via Bluetooth.