It would be great if you could have some fun time before you needed to do any serious wrenching, but it looks like you're going to have to deal with that shifter cable before you can get reliable access to all 3 ratios. In picture 2, I see the steel part of the casing poking past the chain stay mounted fulcrum clip leaving only the outer plastic covering held by the clip. That alone will lead to shifting trouble. As you can see in the above picture, that would never happen with a normal type sleeved fulcrum clip. You could try to patch up this cable but I think you would be best off scrapping that whole shifter cable, buying a new cable, pulley wheel and top tube mounted fulcrum clip and starting from scratch. The modern cables come with a pinch bolt adapter/ barrel adjuster assembly that might not be as elegant as the straight cable to barrel connection, but makes installation much easier since you don't have to worry about the overall cable length. There are ways to eliminate the pinch bolt, but that's a whole other subject.