I wouldn't call it another form of API. It's more like an SDK. If you were accessing a REST API from Android, iOS, Windows, Mac, Firefox, they'd be mostly the same. But an SDK for Android and an SDK for iOS has been built for the platform. Often the SDK encapsulates the official API.
That's a direct answer for (2) too - instead of writing a JS SDK or Swift SDK or whatever, it's an AI SDK and shared across Claude, OpenAI, Groq, and so on.
(3) is exactly related to this. The AI has been trained to run MCPs, viewing them as big labeled buttons in their "mind".
I think you got the questions spot on and the answers right there as well.