can Auto Actions also fire when the user updates a specific field in his profile?

On profile edit? Yes, use the after profile update trigger. Please search as I've provided already how to check if a field at time of profile edit has changed or not using auto actions.

I would like to implement a select box where the user can choose between a few vars. Depending on what he chooses, certain fields are made invisible while others show up.

erilam is correct in this regard. CB Conditional is what you're looking for.

No, I don't think this will work. It's not a step-by-step registration I'm looking for. The select box (and the other fields that depend on it) will be visible in the user profile and for the user only. They're not obligatory and there are four entries in the select box that should perform different actions. Or maybe I got something wrong?

Step By Step is an "after thought" feature in CB Conditional. It's absolutely not what the plugin is about. CB Conditional provides show/hide required/nonrequired of fields based off another fields value dynamically.