Is it obligatory to re-perform the Dhuhr and Asr prayers upon arrival in America, and is any prayer obligatory during a journey in which the time is Dhuhr throughout the entire duration of the travel?

If you pray Dhuhr and Asr at the airport, your prayer is valid, and you are not required to repeat it upon arrival. It is permissible to combine Asr with Dhuhr at the airport if it is outside your city. However, if it is within your city, it is preferable not to combine them, unless delaying Asr would be difficult for you; in that case, it is permissible to combine them by praying Asr with Dhuhr (without shortening), due to the strong presumption that you will not be able to perform it at its proper time. It is also permissible to delay both prayers until you arrive before sunset and pray them combined and shortened if the airport is outside the city where you will be staying.