Are 'with' and 'poor' correctly used in the given sentences?

Are 'with,' 'poor,' and 'health' correctly used in the given sentences?

1. Having a healthy diet starts with understanding the fundamentals of nutrition.
2. Good nutrition starts not with counting calories but with determining what foods to eat and how much to eat.
3. Poor nutrition can result in a number of health problems, including lack of energy, poor sleep, hypertension, obesity, diabetes, and many others.