Is it possible to use Web Audio API server-side (e.g. in Node.js)? I'd like to process "offline" my audio files in order to apply audio effects (http://stackoverflow.com/a/8101383/553341).
Not at the moment, no. You'd basically have to adapt the webkit source code and do all the plumbing yourself.
Have a look at https://github.com/sebpiq/node-web-audio-api
Most of the AudioNodes are not implemented, but in the future it might become useful.