What a grandmaster she could have become had she not chosen an evil path.

@Zerowinger
 
I think all variants you listed are possible. Even existing 6x6 chess variants have different sets of figures, and rules for chess from another cartoon world can bring new variants. https://en.wikipedia.org/wiki/Minichess#6×6_chess
 
Actually I think 6x6 chessboard is used for artistic purposes, to give different feel for another world and simplify everything for cartoon style (same way disney uses 4 fingers or MLP horse hooves are totally very-very simplified real horse hoofs, merging all joints and fur together)

So have Cozy and Tirek already taken out each others’ bishops and queens, or did the animators just not include them?

How appropriate.

@Prince Leaf  
So true. They want to include a chessboard in the shot for whatever reason…but can rarely be bothered to draw it correctly, or use a viable arrangement of pieces (e.g. as having all the bishops on the same colour, pawns on the back row, too many pieces of a given type, etc).
 
Personally, I prefer 3D chess. Or Mega-chess. Or 4D chess. xD I once tried to develop a few of my own variants, with limited success. xD

@Prince Leaf  
They compensated by only making the board 6 rows deep :P

@Yet One More Idiot  
It is so rare for a chessboard and the position to appear normal in any cartoon that I’m more surprised when things actually are correct.

@TheMasterEraser  
And of Deep Blue.
 
…wait, does that mean IBM are currently positioning themselves to take over the world? xD
 
Also, I question the accuracy of that chessboard; the animators could’ve at least made it 8x8 instead of, apparently, 6x6…

Cozy enjoying a game of Los Alamos chess, I see.

It was also the original programmed purpose of the Master Control Program in Tron.