If/when a Pope does something wrong, who is he representing? is he a man who sins, just like the rest of us, or is he supposed to be sort of like the face of God, thereby showing that if he did something wrong, then the pope himself is a fake?
in other words, if a pope did do something unarguably sinful (so not debatable) what would that mean for him specifically, and then Catholics in general?
The Pope is not "the face of God": he is "primus inter pares"1 (the first between equals) - he is not "more bishop" than any other bishop, or "more perfect" than any other man. He is a sinner like all of us, but it's wise and the Church (under Holy Spirit's guide) choose him as chief.
He is not infallible (except in some special occasions, but historically it happened only two times) and he often consult other bishops for his decisions
1 "primus inter partes" is not an official title of the Pope (it refers to the Patriarchs of Constantinople, chief of Orthodox church). The official title of the Pope is "Vicar of Christ", but it's only a semantic difference. The Sacrament of Ordination has only 3 steps, and the pope has the "fullness of the sacrament of Holy Orders" just like other bishops. Pope, Cardinal, Archbishop and similar are human titles, no related to Sacraments - Source: Catechism of the Catholic Church
A great many popes have done things that were less than ideal. I choose one of the Church's worst popes as an example - Pope John XII (955-964), whose pontificate became infamous for the alleged depravity and worldliness with which he conducted it. He was accused of simony and depravity, including adultery and rape, castration, murder and gambling. Wikipedia cites Liudprand of Cremona, who gave an account of the charges levelled against him at the Synod of Rome in 963:
Then, rising up, the cardinal priest Peter testified that he himself had seen John XII celebrate Mass without taking communion. John, bishop of Narni, and John, a cardinal deacon, professed that they themselves saw that a deacon had been ordained in a horse stable, but were unsure of the time. Benedict, cardinal deacon, with other co-deacons and priests, said they knew that he had been paid for ordaining bishops, specifically that he had ordained a ten-year-old bishop in the city of Todi... They testified about his adultery, which they did not see with their own eyes, but nonetheless knew with certainty: he had fornicated with the widow of Rainier, with Stephana his father's concubine, with the widow Anna, and with his own niece, and he made the sacred palace into a whorehouse. They said that he had gone hunting publicly; that he had blinded his confessor Benedict, and thereafter Benedict had died; that he had killed John, cardinal subdeacon, after castrating him; and that he had set fires, girded on a sword, and put on a helmet and cuirass. All, clerics as well as laymen, declared that he had toasted to the devil with wine. They said when playing at dice, he invoked Jupiter, Venus and other demons. They even said he did not celebrate Matins at the canonical hours nor did he make the sign of the cross."
The Catholic Encyclopedia tells us that the Holy Roman Emperor, Otto, eventually convened a synod of bishops to depose John and install Leo VIII, but this ended in failure, with the enthronement of Leo almost universally regarded as invalid.
then the pope himself is a fake? and what would this mean for Catholics in general?
The Church does not consider John a "fake", retaining him in the list of true popes, although it is certainly embarrassed by the heritage of popes such as he. Few Catholics know anything about the wrongs committed by early and medieval popes, so this has no meaning for them.
What about present and future popes?
Modern popes are more restrained, more pious and less worldly than those like John XII, so it is improbable in the extreme that any future pope would bring such dishonour on the Church. Over the last decades, suggestions have been made of minor peccadillos such as nepotism, financial mismanagement and even public homosexuality by one pope or another, but even if these claims are in any way true, the facts have been kept under wraps.
It is said that we all sin, and all must confess and seek forgiveness. When a pope sins, he represents only himself in so doing, and it is incumbent on him to confess and seek forgiveness for his sins. Until 2015, his confessor was Father Berislao Ostojic, but Pope Francis seems to be keeping quiet about the identity of Ostojic's successor.
In order for the Pope to be a Catholic in good standing he too must go to confession at least once a year. In order for the sacrament to be properly ordered according to the Catholic Catechism he must have a cardinal sin to confess. Hence the Pope is like the rest of us, a sinner. The Pope is the visible leader of the church which means he must be the servant of all the faithful. Hence, at the beginning of the Catholic catechism the Popes letter begins: from the servant of the servants of God.
Those who oppose the Roman Catholic Church tend to focus on "bad" popes, bishops, priests, deacons and laity to indicate the failure of the original universal Church. The problem with this line of thinking is that it stands in opposition to Mat 16:18 (NAB). When Jesus first mentions that he will establish a "church" he says "the gates of the netherworld shall never overcome it." To say the church was, at any point, truly overcome by evil is to say Jesus was wrong. We do well to consider that even one of the twelve disciples Jesus chose betrayed him and yet Judas betrayal did not render Jesus' work as invalid. In the same way a Pope that betrays the church, the body of Christ, does not render the church invalid.