Yes. Yes. These are phantom sensations or pain. They are very common soon after an amputation and can persist sometimes years after. Most often they die down with time. People who had a lot of pain just prior to the amputation are more likely to experience phantom pain.
Phantom Limb. It is quite common for amputees to experience sensations on the removed limb. Many times it is pain and we call it phantom pain; sometimes, in your case it can feel different. A study reported in the nejm found that using a mirror to 'visualize' the removed body part actually helped resolve those sensations. http://www.nejm.org/doi/full/10.1056/nejmc071927.
Yes. This is related to phantom limb pain. When the leg is removed, the nerves that run to the feet are cut, with the remaining part still attached inside you. When stimulated, these nerves can send signals that are misinterpreted by the brain as coming from the amputated limb. Itching is sometimes how nerve regeneration feels. Pain management specialists can help if you develop chronic nerve pain.