I'll answer those questions out of order.
I do that sort of thing.
How long depends on the facts of the case, but I'd say an average of six to eight months.
If a state having reciprocal concealed carry privileges has a statute prohibiting felons from being in possession, that statute trumps the out of state permit. So, without more, having obtained a concealed carry permit in Virginia will not give a person any rights in other states. More on this below.
It is not specifically necessary to petition the Governor. One can petition a circuit court for the same relief. The statute says, "Governor or other authority", and a circuit court judge is the repository of the sovereign judicial power of the Commonwealth. I'd recommend that route, unless one has political reasons for trusting the office of the Governor. Perhaps after the next election...
Whether one petitions the Governor or a circuit court, once he receives a favorable outcome, he can petition the circuit court (if you start with the court, you can do it all in one proceeding) for restoration of rights and the granting of the permit. If one has already applied for a concealed carry permit and been denied, the petition proceeding in the circuit court can be combined with the hearing upon the denial.
Ok, here's the interstate and attorney general of the U.S. addendum: full faith and credit. Under the U.S. Constitution, every state is required to give "full faith and credit" to the official acts of every other state, as is the United States. So, if one petitions the circuit court as outlined above and receives a favorable outcome, he can obtain a final order from that court that says his rights have been restored. That final order is an official act of the Commonwealth, entitled to full faith and credit in every other state. It will be accepted automatically by the U.S., with respect to the transfer provisions of the Gun Control Act (make sure a certified copy of the order is sent to the appropriate personnel at the Va. State Police and the Department of Criminal Justice Services so the database thing will be updated). But it will be necessary to file a petition in every state in which one wishes to have his permit honored, to "domesticate" the final order in that state; once that's done, however, that order becomes a final order in the court in which it's filed.