Is it permissible to accept and use a phone gifted by one's father, despite doubting its source and whether it is from lawful money, given the need for it?

Your ignorance of the source of the mobile phone your father gifted you does not affect the permissibility of accepting it. It is not prescribed to inquire about the source of a Muslim's wealth unless it is known that their money is unlawful. The basic principle is that what is in a Muslim's possession belongs to them. As for someone whose money is known to be unlawful, it is not permissible to accept anything from them, whether as a gift or through exchange. If a person's situation is unknown, then the unknown is treated as non-existent. If a man is known to have unlawful money, then dealing with him should be avoided out of piety. However, there is no doubt in dealing with a Muslim whose situation is discreet.