Xbox Play Anywhere, X-Box Gold Needed For Online Multiplayer?

You don't need Gold to play online on Windows 10 PC, this applies only to Xbox Anywhere games purchased from the Store or claimed through a Game pass subscription for PC. Xbox console does require the subscription.

You do need an Xbox Live Gold membership or Game Pass Ultimate membership. Although there is one thing I would like to add: Some online games do not require a Gold membership, for example: Warframe (only solo). And there is many other games that you can
play offline/single player.

Yes you do need a gold subscription or a Xbox Game pass Ultimate subscription to play online multiplayer on the Xbox console. If you plan on to play on PC, then you do not need gold.

For Xbox 360, if you want the second son to be able to play online, you would need a second Gold subscription.

If they just want to play against each other locally, they would need a game which supports local multiplayer (it will say this on the back of the box), but here they would not need a Gold subscription.

If the second son just wants an account, he can have that without needing Gold - he could just have an online account. He would only need Gold if he wants to use it to play online multiplayer, or intends to use it to access Netflix or other Apps.

If you're playing on a PC then you do not need to have a Gold membership. If you're playing on an Xbox One then you do need to have a Gold membership to play online multiplayer games.

As for the other systems, online multiplayer rules and charges will apply as per that platform.