All devices that you have plugged in should show in the Windows Device Manager, whether they have a driver or not. In Windows 7, hit the windows key and type device m in the search and you should see device manager, select that.
In the top level of the device manager, if you plug in the wireless controller, you should see a top level of "Microsoft Common Controller For Windows Class" if you don't see that, and the device is actually working, it should show in an "Other devices" or "Unknown devices" section. If that's the case follow the instructions in this article. Or better yet, this article here which seems to work for people that can't get it going any other way. It says it's for Vista but Vista and Windows 7 share the same driver model.
If you plug it in and nothing happens at all, then you probably have a dead device. I believe the light comes on when it gets power, even before the proper driver is loaded. (not 100% on this) Try it on a second computer.
I had a perfectly good Xbox 360 wireless adapter that worked for a few years, not tons of use. Then I pulled it out of a drawer after it lay dormant for 6 months or so, and plugged into a computer it worked on before, it didn't do anything. I tried a second computer, same thing. The computers didn't even recognize it as a device.
I went out and bought a new one at Best Buy, plugged it in, and it worked, no problem.
No, you cannot connect the Xbox One controller to the 360 receivers. They are different controllers with different tech and mechanical workings inside, so they share 0 compatibility. You can, however, just plug the XB1 controller in using any old micro USB cable (they are common household items nowadays) and it will work.
Best Answer
For this to work with a game, it needs to be part of Microsoft's Xbox Play Anywhere program. That link will take you to the program's official page, with a list of current and upcoming Play Anywhere games. Unfortunately, if the game isn't on that list then you will have to purchase it on your PC to play it there.