You are in Pakistan and want to bring birds into Pakistan? Do I understand that correctly?
You will have to contact whatever agency handles agriculture in Pakistan. They will tell you what the rules are. There are no rules concerning chickens leaving the USA. Nobody cares because they are leaving. It is coming into the new country where they will have to meet conditions.
Birds travel just fine. You get the seller to deliver them to the airport in a proper crate and they should be in Pakistan in no more than 12-14 hours. Most airlines will carry day old chicks, which weigh very little and fit into a small box, so shipping wouldn't be horrendous. the shipping time would be OK for day old chicks.
As to what health papers and if there is a quarantine, you will have to get that information from the government of Pakistan.