I work as a temporary associate for a company in VA. Can I work through my 1 hour lunch to make up time missed during the same week? Which means that I would work 9 hours in a day. Also, can I work through my 1 hour lunch and take 2 hours the next day?

There are no laws mandating breaks or lunches in your state. Working out such a schedule is strictly between you and your employer. However keep in mind that if this puts you in excess 40 hours a week your employer will be forced to pay overtime.