The Character Sheet is public, just as you can see someone's Character Sheet they can see yours. I did not code it to be private.
If by "Elsewhere" You mean in other Role-playing Games here, then yes, some of the other Role-playing Games in the Community use different versions of the Character Sheet. If by "Elsewhere" You mean on other sites then you can but I would ask that you create it via the support Thread: Text Role-playing Game Character Creator and not in a current Role-playing Game .
What controls the background color of our character's avatar? I noticed mine has a dark shade and one player has a lighter color than the rest.
That's a new feature of the Text Role-playing Game Creator. One of the new fields added was "Character Status". Depending on your choice it displays a color and title in your Posts. Read more: here in the first Post under "Integration With Character Posts".
Just wanting to know if I'm putting my foot in my mouth. Is the player who is chosen as party leader chosen by vote of the players or by the Dungeon Master?
Players generally choose their leader but the party leader system on a whole is a choice of the Dungeon Master / GM as some of them like for each Player to state their Character's actions within the same Post they do.
Dividing It Up
As this Board gets more and more saturated with Dungeons & Dragons / Pathfinder I will have to add special buttons to the top to take Players to their section / version of the game so they aren't confused with all the different active Threads. This is especially true for new Players who are easily distracted and tend to give up quickly. An example of this is how I have a special section within the Science Fiction Board for Traveller alone: Traveller Role-playing Game. Therefore, Dungeons & Dragons 5e, AD&D, Dungeons & Dragons 3.5 will all have their own sections.