Occasionally the Donor Forum's member will hold contests such as this, but it's not that common.
Although there is no dedicated forum for contests, you are welcome to hold certain contests in My Events, General Chat, or Donor Forums, so long as you're not giving it away for free, or for hitting a specific post number.
Also, you can have graphic arts based contests in the General Design forum.
I would suggest reading the sticking to properly format your suggestion, but I like your suggestion for sure.
The only problem with the "answer and receive FG automatically" is trolls would ask dumb questions that make no sense, or input answers that are irrelevant to the question just to be trolls.