Is the imamate of one who experienced a minor ritual impurity (passing wind) while urinating valid, if he then performed ablution (wudu') and led the people in prayer, knowing that he only cleaned the front private part with a wet tissue and did not clean the back private part after the wind was expelled?

What that person did by cleaning himself with a wet tissue to remove urine is not correct. It is better to clean oneself with water, or with a dry tissue, because a wet tissue is feared to spread impurity. As for cleaning oneself after passing wind, it is neither obligatory nor prescribed at all. Therefore, his prayer is valid, and his leading of prayer is valid, and his colleague erred by not praying behind him.