Is it permissible to have the intention, when performing acts of obedience, of encouraging and motivating one's family to do likewise, and will a person be rewarded for refraining from unlawful acts solely out of fear of punishment?

Having multiple intentions in good deeds is a good thing and increases the reward, as the Prophet, peace and blessings be upon him, said: "Indeed, actions are by intentions." Abandoning sins out of fear of Allah is a good deed that is rewarded, and the reward increases as the number of good intentions for avoiding the forbidden multiplies. The best intentions are those that combine worship of Allah, obedience to His command, fear of Him, love for Him, and hope in Him.