If I buy the game on the battle net client, I will own it on a Xbox console when I link the account?
No. You have to buy it on each platform you want to play it on.
The answer is no. These are different platforms, similar to how games purchased on Steam can’t be accessed through the Battle.net client, since they’re on separate marketplaces.
If you’re looking to play on both Console and PC, the best option is to subscribe to Game Pass Ultimate without buying the base game. You’ll need a subscription to play multiplayer on Console, so Game Pass Ultimate offers better value. However, for the DLC, you’ll need to purchase it separately on each platform.