Does what I eat affect my skin?

Yes. Saturated fat and carbohydrates stimulate production of cytokines from fat cells that destroy collagen. The destruction accelerates with age and the amount of fat and carbs consumed. It can make some skin cells grow causing thick, dark skin and skin tags.
Not much. Your diet is very important for other reasons. Unless you suffer severe vitamin deficiencies, rare in usa, your skin is not a mirror of your diet.