靴 (화) is correct. I don't know where the lesson notes get 鞋 (not 화 but 혜) from. It seems to mean sandals (샌들).
Since it is sock's' and sneaker's', it would be grammatically correct to say 양말'들'과 운동화'들'.
Yes, but it sounds very unnatural. You would use plurals when you are speaking of specific socks or sneakers. Even then it sounds a bit weird.
why not ~하고?