To answer your questions in order:
  1. Yes most definitely have writing contests. I’ve only been on SN for a few days and I’m already impressed by the creativity & ideas I’ve come across.
  2. In a past experience of writing I’ve seen contests with a very short timeframe and using 2 or 3 words as starting points. As has already been mentioned these could be something that links the SN community. A monthly or bi-monthly schedule would be a good starting point to build from.