If charges are filed and the first court date is after the statute of limitation, is that a failure to prosecute me in a timely fashion?

No. All that matters is that the Prosecutor filed charges before the statute of limitations expired. It doesn't matter that they got your last name wrong. The Prosecutor or the police officer can correct any errors at any time, up to and including the first day of trial. That is not enough to be able to get the charges dismissed.