In my last comment to Josh's answer I have announced that I wanted to set up an app that would read the site feed and post the questions to Facebook.
I have now set up that app, and the first three posts have been posted. You can see them here: https://www.facebook.com/CogSciStackExchange
The app allows some settings, and if you have an opinion on how we should set them, I'm happy to hear it.
How often do we want to post updates?
The shortest span is "no faster than" every half hour, the default setting was four times a day. I have set this to every half hour, because as long as we don't spam our followers (and there is currently no danger of that), it will be nice for them to be able to see the posts and react to them quickly. But see no. 3.
How many questions do we maximally want to post per update?
At the moment I have set this to unlimited, since we don’t have that many new questions per day.
Do we want to post the questions as quickly as possible, or do we want to wait for some (sometimes necessary) edits before we post them on FB?
At the moment I have set this to wait 24 hours before a question is posted. This should give some minimum time for someone to edit or flag most questions that need to be edited or closed. On the other hand (see no. 1), it would be nice to have little delay.
Do we want the full text of the question posted, and if not, how long/short (in characters) should the quote be?
I have currently set this to full text.
Do we want the text of the posts to be published over the grey box (and over the link to the question or inside the box (and below the link)?
As you can see, the text is automatically cut off if the question is posted inside the box. I have currently set this to over the box.