I have multiple players in my console . If I buy one Xbox Gold can all the players on my...

If you set the console as your Home, you can share the Gold with anyone else that uses the console. https://support.xbox.com/en-US/help/hardware-network/console/my-home-xbox

If he logs in on your Xbox One, all other players can also use Xbox Live Gold features.

Or if he set your Xbox One to HOME console. Then he does not be logged in. All players have access to his games and gold services.

BUT if he goes to other Xbox One console and set it HOME console. Then you do not have access anymore to gold services.

support.xbox.com/.../my-home-xbox

So, if four players (two at my house and two at a neighbor's house) are playing each other, all 4 players can have and use headsets at the same time? Sorry, I just want to be clear on this because my sons are leaving my wife and I quite confused about it.
I don't want to buy multiple headsets if they can't be used.

I just bought a used xbox 360. If I get a gold membership, can all profiles one my console share it? How do I set that up?

Although, you can play with the Player Guest option. Not the same, but it is something.

I currently own 4 360s, each with Gold, each in a different room, for myself and my 3 kids. I was planning on buying that same number of Xbox One consoles. However, they do not have room for a Kinect in their rooms. While I understand that we can all use
the first install on my console, I'm not crazy about having to pay again for them to play in their room.

If I can't use multiple consoles as I do now, I won't be buying a single Xbox One, much less 4.