Node.js Express for REST? Are there controllers in Express?

Have you seen Swagger? An API server available in Node.js with an automated UI Generator.

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To answer your primary question, yes there are controllers in Express. They are more akin to Sinatra (express is modeled after Sinatra rather than Rails). By that I mean they are light-weight method mappings to routes you define.

Browser: jQuery, maybe JavaScriptMVC (there are some things I don't like about JavaScriptMVC (like the routes), maybe I write my own little MVC or do you know a better suited framework?

I'd go with spine.js or backbone.js personally.

Express looks really nice but I'm missing MVC. I couldn't find any project that implements mvc and is build on Express. Is there a reason for that? Routing is nice in Express but I need a way to spread code across multiple files (controllers would be the best way I guess, the application won't be small and I need it maintainable)

You can spread files out all you want. I answered this here.

The application will be more or less completely based on AJAX (json) requests. Is Express the right framework for such applications, anyway? I think the best way to write this project is to expose a json REST api which can then be queried by the web application over AJAX and by a mobile device app (which I'm also going to write). In my opinion Express' route system is quite suited for REST. But feel free to recommend other frameworks.

I haven't built a 1 page app yet in node but from what I can tell, almost everyone seems to be using socket.io with backbone. That's not to say you can't, just that you'll find more examples that way.