Which citizen-science astrophysics and astronomy projects currently exist? Most of you probably know of Galaxy Zoo, a citizen-science project to classify galaxies with help of untrained hobby-researchers. In mathematics, the Polymath Project is well known for its success. SETI@home, for example, is not citizen science as you do no active research, only your computer is crowdsourced.
Are there projects requiring more than simply visual classification of astronomical pictures?
 A: The project which I know the best is Citizen Sky, organized by the American Association of Variable Star Observers, which has been encouraging amateur science for over a century. Highly recommended!
A: There are several other astronomy citizen science projects1, most of them through the Zooniverse.  These include the Galaxy Zoo you mentioned, but also Galaxy Zoo Hubble, Ice Hunters, Planet Hunters, The Milky Way Project, Solar Stormwatch, Galaxy Zoo Mergers, and Galaxy Zoo Supernovae (and two others in the planning stages that aren't public yet1).  Another that I know of is the Stardust @Home that asks you to look through images to find potential dust grains.
That answers your first non-question question.  As for your first question-question, by their nature, citizen science projects do not generally ask you to do anything more complicated than very basic visual classification.  There are several reasons for this.  The two main ones are (1) humans are still significantly better at pattern recognition of this type than computers (otherwise we'd have computers do it), and (2) it requires significantly more training and then subsequent validation by scientists to have non-experts do more complicated tasks.
That said, I can tell you we1 are working on some more advanced visual classification for the Moon Zoo project where we're thinking of implementing a user "level" system similar to video games.  This would work, for example, by having beginning users go through a quick tutorial to train them how to identify craters.  After they have done a certain number and they have been verified against experts' classifications, then the user could advance to a "Level 2" where they would be asked to do something more complicated, like correct an automated algorithm's detection of craters on an image.  "Level 3" may ask them to identify more interesting features of, well, interest, like odd-looking craters, bright rays, elliptical craters.  A "Level 4" could be adding in identification of linear features on different images.  With each new level, we would need to have the user go through more training and need to further validate their work for the larger scientific community to accept it (scientists can be a bit snobbish and it takes some convincing to prove these are viable).
So we're working on it, but really it's the basic "grunt work" of visual analysis that these kinds of projects are best suited for at the moment.
1Full-disclosure: I was recently awarded a NASA postdoctoral fellowship for citizen science project management, and I am currently the P.I. on Moon Zoo.
