This was a beautiful puzzle, but I did not receive as many answers as I would have liked. Correct answers came from Pratik Poddar and Suman Saraf. My younger daughter, Arushi (9), solved it with some help.
Answer to all the variations is “No”. Here is the explanation – look at the figure below:
I took the liberty of coloring the same grid in a chess like pattern. Notice that in any move, a penny on the black square can only move to another square with black color, and vice versa, one on white can only move to a white one. Also notice that even if a diagonal move was allowed, this principle remains true. Above the line, we have pennies on 9 black squares and 6 white squares. On the other side, there are 6 black and 9 white squares. Therefore 9 pennies from black squares above the line cannot find enough black squares below the line.
The above is true even if we remove one penny from a black square or even 2 pennies from a black square.
Hope you all enjoyed the puzzle!