⟩ Can we use jQuery with Node.js?
No. It's going to be quite a big effort to port a browser environment to node.
Another approach, that I'm currently investigating for unit testing, is to create "Mock" version of jQuery that provides callbacks whenever a selector is called.
This way you could unit test your jQuery plugins without actually having a DOM. You'll still have to test in real browsers to see if your code works in the wild, but if you discover browser specific issues, you can easily "mock" those in your unit tests as well.
I'll push something to github.com/felixge once it's ready to show.