Is it permissible to combine the intention of drawing closer to Allah with the intention of losing weight when observing a voluntary fast? And will a person be rewarded in the Hereafter if they avoid sins out of fear of what people might say, or of public disgrace, or of dishonor, or out of chivalry and self-respect?

There is no harm in combining the intention of reward for fasting with the intention of weight loss, even though it is preferable to intend only the reward. Weight loss occurs through fasting whether it is intended or not. Al-Suyuti mentioned that combining the intention of worship with something that is not worship does not invalidate it, as long as the other matter is achieved by the occurrence of the worship, such as intending to cool down with ablution and ghusl (definition), and intending diet and treatment with fasting. As for abandoning sin with the intention of fear of people or shyness from them, this has been detailed previously.