Does prayer become invalidated upon smelling a foul odor during it and being unable to swallow one's saliva, and if it is invalidated, how are the missed prayers to be made up given that their number is not remembered?

The presence of something in a worshipper's mouth does not invalidate the prayer, unless it prevents him from articulating the letters from their proper points of articulation. If it does not prevent him, then the prayer is valid and he does not need to repeat it. The feeling that an odor has mingled with the saliva might be a result of waswasah (whisperings of Satan), and one should not pay attention to it.