Is residence and the money resulting from it in Sweden unlawful due to lying and fabricating information to obtain them, and is it permissible to falsify one's age to join one's father there?

Your father's undertaking of seeking political asylum without being persecuted in his country entails Sharia prohibitions: lying, taking money through deception, and residing in disbelieving countries, which may lead to neglecting duties and falling into prohibitions, in addition to falsifying age. Your father must repent to Allah for these forbidden acts, and if you can stay in a Muslim country with a Mahram, then do so to avoid these prohibitions.