For cylindrical, definitely if you have the settings right to match the cylinder that you're mapping onto, the checkerboard should indeed wrap around the cylinder and look even. For a unit half-cylinder centered with its bottom at 0, 0, 0, I believe the parameters would be 0 0 0 0 -1 0 0 0 1 180. (I color coded the numbers to make it easier to read.) With those settings, the left edge of the texture would map to left edge of the half-cylinder, the right edge of the texture would map to the right edge of the half-cylinder, and the middle bottom of the texture would be at coordinates 0, 0, 1, which would be the bottom center of the half-cylinder's surface.
For spherical, I'm not completely sure what it's supposed to look like, but if you used the same texture map for all three shapes, I don't think what you have is right. However, I believe for a unit sphere at the origin with texmap params 0 0 0 0 0 1 1 0 1 180 180, the center of the texture would map to the point at 0 0 1, and the texture near that point would look fairly non-warped. The checkerboard would get progressively more warped as you move further from that point. The centers of each of the four edges of the texture would map to the edge of the half-sphere, and everything outside a circle inscribed in the texture's square would be invisible (out of bounds).
For spherical, I'm not completely sure what it's supposed to look like, but if you used the same texture map for all three shapes, I don't think what you have is right. However, I believe for a unit sphere at the origin with texmap params 0 0 0 0 0 1 1 0 1 180 180, the center of the texture would map to the point at 0 0 1, and the texture near that point would look fairly non-warped. The checkerboard would get progressively more warped as you move further from that point. The centers of each of the four edges of the texture would map to the edge of the half-sphere, and everything outside a circle inscribed in the texture's square would be invisible (out of bounds).