I was told that if I don't eat enough it will affect my milk production. Is this true?

Partially true. When nursing make sure to eat well and drink enough fluids. This will keep you healthy. Your milk will be of good quality and amount even if you don't eat well. In reality the most influential factor in how much milk you produce is your baby nursing as frequently as she needs. If you introduce formula to supplement, your milk will start to become insufficient.
Yes and no. If you eat a balanced diet based on your pre-pregnancy weight and consume adequate fluids you will be fine. In fact you can shed up to one pound every 10 days while your body makes great milk. If you get to your desired weight, add more calories. Keep taking your prenatal vitamins. If you eat "extra" to make milk you can bet the pregnancy weight will stay.