Definitely depends on the airline. The big US carriers all have their own systems, but the vast majority of foreign carriers use a GDS on their backend. It's just not worth building in-house unless you're at very large scale.
Old non-US airlines definitely have their own systems. I know Scandinavian Airlines have their own, and it looks like Lufthansa and KLM-Air France do too.[0]
I would be surprised if say British Airways didn't.
Nope, they all use Amadeus, except for Scandinavian Airlines. Actually Lufthansa and Air France CREATED Amadeus to consolidate their GDS operations. They're now pissed because Amadeus takes a piece of the cake and plays weird pricing shenanigans, but they're still there.
These guys do use Amadeus on the backend to manage their ticket inventory. Specifically Amadeus Altea - though you are correct that they try very hard to avoid paying GDS distribution fees for selling tickets through the GDS third party sales channels (which is what that article is discussing).
