I think the best design is going to be chosen based on the Ember app’s dependency on a REST API.
If you’re building an app that’s 100% on the client and needs absolutely no server connection, then it would be crazy to pick something that depended on a Rails-like API to exist.
But if you’re building an app that is more-traditional in the client/server space, then it may be advantageous to send email from the server, initiated by a PUT/POST from your Ember client.
When it comes to transport, pick what best suits your application. If you need a realtime system, then go websockets. If you don’t and good ol’ HTTP request/response meets your needs, go with that.
I’d say it’s best practice to let Ember just represent your data. Keep all your business logic and complex search stuff within the bounds of your API server. If you’re building an email client, it’s easy to push down JSON to represent a list of messages, folders and search.
Hope this helps.