Is it permissible to obtain a forged nationality from a grandfather to facilitate matters of study, livelihood, and marriage, and to improve financial status, given the difficult circumstances the questioner and his family are facing?

Forgery is forbidden in Islamic law. It includes deception in documents and imitating handwriting with the intention of defrauding. What you mentioned does not permit you this prohibited act for three reasons: First: Because it is prohibited forgery and deception. Second: Because it is attributing oneself to other than one's father, and it is stated in the Hadith: "Whoever attributes himself to other than his father – while knowing he is not his father – then Paradise is forbidden for him." Third: Because it violates the laws of the state that granted you residency under conditions that prevent deceiving and defrauding it.