Wouldn’t every registered republican have standing?
Not using the Obama eligibility test.
No. The only people with standing are those people who, under state or federal law, have the right to contest a candidate's qualifications to appear on the ballot either in a primary or general election.
No. To have standing in Federal court, you must show that you would suffer a particular harm specific to you. In this case, since the "harm" alleged would be generalized to the public at large, or to all Republicans, you could not get standing as an individual voter. I am not sure who could actually get standing in Federal court - Congress probably, perhaps the electoral college, or possibly another presidential candidate. In state court that would be a different issue, and could vary from state to state depending upon the specific election laws.
Of course, one of the peculiar aspects of how we elect our president makes it very difficult to adjudicate this topic. That is because you don't actually vote for president. In the primary you vote for delegates who are pledged to vote to nominate a specific person; in the general, you vote for electors pledged to vote for your candidate. But in neither case are you actually voting directly for the candidate.