Yes I use a wired one for wwe2k16

Yes, both the XB360 and XB1 controllers can be used on games with "controller support".

A bit more technical information:

Both the listed controllers (and the Steam controller) use an API called "XInput". Most games these days support XInput controllers (as Microsoft has made it easy to support it in Windows games -- thus they've become a de facto standard on Windows). Microsoft also includes drivers for the XB360 and XB1 controllers for most recent versions of Windows.