Is a Muslim who changes his religion "on paper" in order to obtain residency in a foreign country considered to have left Islam, or does he remain a Muslim, taking into account that he still believes in his heart and performs his religious obligations? And what should he do in this situation?

Changing one's religion in official documents from Muslim to another faith constitutes disbelief (kufr) and apostasy from Islam, even if the person claims their heart is firm in faith. This is unless there is compelling coercion, such as a genuine threat of death. Obtaining residency or citizenship is not an excuse. Indeed, one who utters disbelief in jest commits disbelief, so how much more so one who records it in official documents? Therefore, whoever has done this must make sincere repentance and hasten to change their religion in official documents back to Islam, and seek a country that does not require disbelief to obtain residency.