Is it possible to feel pain in some part of a body, but the pain "feeling" is introduced somewhere else?
Yes, this is pretty common.  Examples include 


sciatica, pain felt down the back of a leg to the foot, from irritation to components of the sciatic nerve but commonly at the level of the sciatic nerve roots
angina pectoris, pain from myocardial ischaemia felt in the throat (Latin angina "infection of the throat"), arms, chest etc
shoulder tip pain from a subphrenic abscess


The clearest possible example of this is phantom limb pain, where pain is perceived to be arising from an amputated limb indicating that the pain is arising from central mechanisms.