You can run any Javascript code on both server and client, including Ember. There are definitely people using Ember that way.
Ember actually offers two different levels of server-side runnability. There is a dedicated, DOM-less build of the Ember core runtime that makes it easy to execute all your application & model logic on the server side, without depending on a browser-like environment. Alternatively, you can use the full Ember build on the server side with a DOM implementation like PhantomJS to render complete pages.
Regardless of the framework you choose, the general tooling & ecosystem support for managing dependencies across both server and client is pretty poor. I use Browserify and it’s ok, but it is far from mature software and it took a lot of tweaking to get everything (including various 3rd-party dependencies) to play nicely together.
If you’re “new to JS and web stuff”, you may want to start with a more typical Ember or Backbone app that is all rendered on the client, and worry later about moving some rendering to the server if you need to. I’m not convinced that there’s a major performance reason to do it, most of the time. And one of the reasons Rendr exists (supporting deep-links) doesn’t apply to Ember, because Ember apps always support deep-links, even when rendering everything in the browser.