The evaluation of results is one of the hardest task in photogrammetry. Therefore questions are: How accurate do you need to be? Are we talking about about accuracies of 1ppm or 1:1,000? How reliable is your hardware for your goal?
1) The reprojection errors do not really yield anything reliable. It just tells you how the chosen function fits into the measurements (is also often referred as internal accuracy). So if your measurements are garbage the result protocol will happily tell you how well it could fit into your garbage. A reliable evaluation is only possible if you have enough external references to get a good approximation for the external accuracy. This can be achieved with precise known distances between targets which have been not included in the calibration step to scale the systems. For a solid calibration with a plane calibration body you'll need six of them. Two as a cross on the main diagonal and four on each side.
2) How big are the circles in the image? You might need to correct your image measurements for circle eccentricity before starting your calibration. Is your measurement volume two dimensional? Only in that case a two dimensional calibration field is a good choice. Circle targets are (at the moment) with a huge distance the most reliable,robust and precise targets. Chessboard targets are mostly used in robotics or computer vision but not really when you expect some level of precision. Also the cx, cy approach is a bad choice if you want to achieve some level of precision since it's arbitrary and has no physical basis. Look for a physical equation like the Brown approach to describe your lens. The parameters are mostly referred as: c (focal length), x0,y0 (principal point) ,r0,A1,A2,A3 (radial symmetric distortion),B1,B2 (radial asymmetric distortion) ,C1,C2 (affine distortion).
