Is it preferable to slaughter (an animal) over giving money when giving charity, and is the reward diminished if the charity is given at the mosque for the officials to distribute, compared to distributing it personally?

The best form of charity is that which provides greater benefit to the poor and needy. So, if their need for meat is greater, then sacrificing an animal and donating the meat is better. If their need for money is greater, then donating money is better. Giving charity at the mosque for distribution does not diminish the reward, provided there is no intention of showing off or seeking reputation. In fact, it might be better because it is closer to concealment, which is preferred in voluntary charity and all other voluntary acts of worship, in accordance with the Almighty’s saying: "If you disclose your charities, it is well; but if you conceal them and give them to the poor, it is better for you."