Most of your questions can share the same answer: every robot is different and there are many ways to tackle different obstacles, so you will have to determine what will work for your robot. For example, I cannot say for sure how to change your P and D coefficients to increase the performance of your robot, but it will likely take quite a bit of trial and error to get it right. Decreasing the P coefficient will make it less responsive to the error proportionally, but increasing the D coefficient will increase its response to changes in the error.
As far as your sensor choice, we generally recommend the digital versions of our QTR reflectance sensors over the analog ones, but both will work. You can find more details of the benefits of using the digital ones from my post in this thread. With either version of the sensor, ambient lighting can cause variations in performance (e.g. fluorescent lights or large windows letting in sunlight can be tricky for the sensors to handle). It is usually a good idea to test and tune your robot in as close to the same environment as you will be competing in, but shielding the sensors could help alleviate as well. Also, note that the sensors measure the IR light reflected off the surface, so if your printed course did not use the same material, that would likely cause some differences.
Brandon