Is it permissible to make use of a mobile phone and a camera previously gifted by an ex-fiancée, and can the mobile phone be gifted to one's father for use in Skype calls and the camera be used for producing da‘wah (preaching) videos?

It is not legislated to give gifts on birthdays, and it is better not to accept them for those who do not fear harm. However, there is no sin upon one who accepts them and benefits from them, for a gift is not forbidden. If the gift is to encourage you towards marriage, then there is no sin in benefiting from it to strengthen affection between the engaged couple, due to the Prophet's (peace be upon him) saying: "Exchange gifts; you will love each other." This is with the avoidance of doing so on innovated occasions.