There's a heck of a lot of carrier pigeons and the like in private hands in the country - they could be confiscated.
We don't (to my knowledge) have a stock of carrier pigeons anymore. Probably still have some, I know hobbyists train them still, but overall not a lot to cover the mass communication we need.
Nope. Mail wouldn't be running - until things had settled enough for someone to start it up again.
Until Martial Law was in place, would the postmen even be working? (Given that everything's going to Hell in a handbasket, I probably wouldn't show up for work, either!) so it would be government mandated couriers on bicycles. Even then, there's still only so much ground that can be covered. It would take weeks to re-establish proper communications.