The sort of head with a blindfold is an orc head, they are the same image as used on the flags for each Barbarian Civ, just get hard to see
Sun Mana Icon means you have Light of Lugus effect (all invisible units in your territory are visible to you), Shadow Mana Icon means you have Chosen of Esus effect (all your units not in cities in your territory are invisible), and Nature Mana icon means you have Dowsing active (can see all resources on the map)
What does Republic actually do? I have been running the civic for quite sometime, and have not seen any Elections taking place. How often are they supposed to happen?
Not nearly often enough is the best answer I can give you on that one. As to what they do:
Each election has a choice between two options. Your first choice has an 80% chance of the first option and a 20% chance of nothing happening. Your second choice is the same for the second option, and the third choice is a 50% chance of either. Most often, you get a choice between two traits (Aggressive or Defender is the one I've gotten the most, but also seen Spiritual, Organized, Industrious, and Philosophical have also shown up), but occasionally you get a choice between Great People.
That's been my experience so far, there may be other things I haven't encountered yet.
Thanks, in some ways I feel that Republic is lacking when those elections don't happen.