Constructing a Pixel Image from Rectangles and Row or Column Rotations
Yes.  For every image, there exists some sequence of operations that generates that image.

I'll describe how to implement a set (x,y) operation, which sets the pixel in the xth row and yth column (and leaves all other pixels unchanged).  It should be clear that you can construct any image you want by repeatedly applying this primitive.

Here's how to implement the "set (x,y)" operation using the 3 operations you gave us:


Rotate column y upwards by x-1 positions, so that position (x,y) moves to position (x,1).
Rotate row 1 leftwards by y-1 positions, so that position (x,1) moves to position (1,1).
Set the pixel in the upper-left corner (the pixel at position (1,1)).
Rotate row 1 rightwards by y-1 positions, so that position (1,1) moves back to (x,1).
Rotate column y downwards by x-1 positions, so that position (x,1) moves to position (x,y).


It is easy to see that each of these 5 steps can be done using one of your three operations, and that afterwards, the pixel at (x,y) is set and all other pixels are left unchanged.

This shows that given a image with dimensions mxn, we can create it using at most 5mn operations.  Of course, this sequence of operations probably isn't the shortest possible.  I don't know whether there is an efficient algorithm to find the shortest sequence of operations to generate a target image.