Is the possible values for a binomial random variable the whole numbers from zero to infinity?

No. A binary distribution gives the probability of obtaining x successes in n trials, x can not be greater than n. The binomial distribution can be used for any whole number greater than 0, but the distribution is not unbounded. So let's say the binomial can determine the probability of x success in n trials where n can be from zero to some finite number less than infinity.