I will add and correct the previous answer.
Will come back. The member, for the most part and "moving", part consists of spongy tissue, which has the properties to expand and thicken, taking in blood. Plus, an erection is not just a rush of blood in x_y, it is the release of hormones and other things that speed up the process. As your heart stops after death, your erect penis will stop receiving blood flow and your erection will stop. Men do not need to explain how sometimes this happens quickly "alive", under certain circumstances. So it does not matter in what position you lie dead, on your stomach or on your back, the muscles of the penis will not have time to stiffen enough so that it remains in a more or less standing state.
Yes, at least because the heart stops and the blood stops circulating. Accordingly, there will be an outflow of blood from the penis, it will accumulate under the influence of gravity at the points of contact of the body with the ground (any other surface on which it will lie), which subsequently forms bruises. Provided that the body lies on its stomach, I find it difficult to answer. I can only assume that the erection will not be so strong, but it will remain for a while. Although there will be no blood pressure, so even before the body hardens, the penis will fall, IMHO.