Nobody teaches the kitten to use the litter box. This it does instinctively. It is in the cats nature to dig a hole, defecate and cover it. Regarding your dog, I suggest you keep on taking him for the regular walks. I was surprised to see my cat eating grass.
Cats do eat grass.
Grass contains folic acid which is an important vitamin supplement to their regular meat diet. There is nothing wrong with your cat. I am planning to have my female cat spayed. If you do not intend to breed your cat, it is a good idea to have her spayed. Cystic ovaries are a common cause for repeated heat cycles which can be truly irritating for you.