What is the ruling on traveling to America for the purpose of obtaining citizenship and studying, then moving to live in a Muslim country? And does swearing an oath to respect the laws of the state and show allegiance to it, without heartfelt consent, in order to obtain citizenship, constitute apostasy from the religion?

It is not permissible to obtain citizenship of a disbelieving country except out of necessity or an overriding legitimate interest, such as when a Muslim is persecuted in their own country and finds no refuge in a Muslim land. If this is the case, and the Muslim finds no Muslim country to welcome them, and is able to preserve their religion, then there is no harm in obtaining American citizenship with the intention of temporary residency until they can reside in a Muslim country.

As for the oath, it is permissible when there is a necessity that allows for obtaining citizenship, and the Muslim should intend to limit their loyalty to the Muslims among them, and strive for the welfare of the country by guiding its people.