Do members of an organization set up as a 501(c)(3) have the ability to nominate and vote for the members of the Board of Directors? If they do are these rights guaranteed by law?

Such organizations are set up according to state law, normally the corporations code pertaining to non-profit corporations. So, in general, it isn't a matter of federal tax law, but rather state law. In some states it is possible to create a no-member non-profit corporation. Check your by-laws for the most direct answer to your question.