For a 5 x 1 rectangle, the maximum sum is 25/7 = 3.571428... This value is attained in the following 29 x 25 symmetric grid (only the upper left quadrant given here):
2 -3 1 1 -2 2 1 2 0 0 0 1 -3 0 0 -2 0 -1 0 0 -3 2 1 0 0 1 2 1 -1 0 0 2 0 0 0 0 -4 0 -2 0 1 1 2 0 0 0 0 1 0 0 0 0 -2 2 0 3 -3 1 0 0 1 0 0 0 0 1 1 0 -1 0 2 -4 2 -1 -2 0 2 0 2 -1 0 0 0 0 0 2 2 -5 -2 0 2 0 0 -1 1 0 1 0 -2 -2 -19 28 0 -6 0 -4 1 2 2 -5 -6 -2 0 5 4 7 -28 28 -6 0 0 -2 -3 -1 6 4 -2 -2 -3 3 7 -3 -22 19 0 0 0 0 0 0 2 -1 0 -1 -2 6 0 -3 -8 10 0 0 0 0 2 0 1 0 0 -2 3 -7 3 -2 0 -1 0 0 -1 0 2 -2 2 -3 0 8 22 -22 -6 0 2 -2 0 0 -4 -6 6 -4 8 8 -28 -28 28 22 0
The 5x1 rectangle is '28 22 0 22 28', with sum 100. All squares have -28 <= sum <= 28.
For a 6x1 rectangle, the maximum sum is 85/23 = 3.695652... which I think is attained in a 36x31 grid. My program and I will try to generate this grid tonight.
For a 7x1 rectangle, the maximum sum is 11/3 = 3.666666..., which is less than the 6x1 rectangle.
I will post more results as they come in.