Can you verify that an estimated coefficient is in fact unbiased? Why?

No, it is not possible to verify that an estimated coefficient is in fact unbiased. The only way to verify the unbiasedness of a coefficient is to know the true underlying population and data generating process, which is impossible. By definition, an unbiased estimator is one which yields a value that is equal to the true value of the parameter being estimated, but this requires knowledge of the true population parameters, which is not usually available.