I suppose you could argue it may have helped keep the boat afloat before they launched Zen, but as far as I know the margins on their game console chips are terrible.
The margins are terrible, but they make it up in volume. :P Seriously though, having a volume product that covers its costs keeps the business running; even if it doesn't make a lot of margin, it enables producing products that do have reasonable margins. Zen, especially since Zen 2, can probably pay for itself now, but it's still nice to have two high volume customers.
Not really. It's only the Xbox and Xbox One, PS4/5 where AMD CPU got used. ATI has some more design wins. Earlier generations were somewhat exotic in comparison to PC.