What should I do about my friend whom I led in Dhuhr prayer while I was without Wudu, and I have already redone my prayer at home? Is it obligatory for me to inform him to redo his prayer, or not?

What you did by re-performing the prayer that you had offered without ablution is what must be done, because the first prayer was not valid due to the absence of purity (tahara), which is a condition for the validity of prayer, based on the Prophet’s (peace and blessings be upon him) saying: "Allah does not accept any prayer without purification." The Muslim Ummah has unanimously agreed that whoever prays in a state of ritual impurity (hadath) while being able to perform ablution, their prayer is invalid and must be repeated. As for your friend who prayed behind you, his prayer is valid, and you are not required to inform him of this.