Hi Everett,
Yes, DirectRT can do that (see the "cursor" sample) but try to avoid using it for a single trial from within MediaLab. It's awkward to keep flipping back and forth on a trial by trial basis. You could do it in MediaLab as a custom item. I'm sure that you could find some simple (depending on your experience) javascript that could record and post where a use clicks on a picture.
And yes, a multiple response item with 13 or more options could be done in HTML. I've attached a sample demographics questionnaire includeds a multiple response item (race, check all that apply). You could take that section and just expand it as necessary. Hope that helps--let me know if you run into any trouble with it!