Ajax is merely a client JS interface to HTTP. A protocol that already exist. It's not a protocol being imposed on web servers.

When you want to impose a new protocol on web servers and browsers, it better not be to solve push notifications only. And that's what SPDY did.

WebSocket isn't just to solve push notifications. It's also for realtime communication, for example chat or multiplayer games. We do have server-sent updates if you just want push notifications.

yes, without compression. Everything you said is covered in SPDY. But SPDY takes a step further and improves the current state of HTTP as well to better transfer documents.