How has apostolic succession been "unbroken" in the wake of depositions, antipopes, immorality, and simony?

Here's what the First Vatican Council says regarding the perpetuity of the Petrine office (ch. 2 "On the Perpetuity of the Primacy of blessed Peter in the Roman Pontiffs"):

That which the Prince of Shepherds and great Shepherd of the sheep, Jesus Christ our Lord, established in the person of the blessed Apostle Peter to secure the perpetual welfare and lasting good of the Church, must, by the same institution, necessarily remain unceasingly in the Church; which, being founded upon the Rock, will stand firm to the end of the world.

Notice, it did not say: "There will be an occupant of the chair of Peter at all times." Perpetuity refers to the institution of the Papacy, not the holders of the office. Otherwise, even interregnums (i.e., the period of time between the death of a Pope and the election of his successor) would "break" the Apostolic succession.

It's also entirely possible and in-line with Catholic theology that Pope Pius XII was the last valid Pope, as sedevacantists* today believe. Even if the world ended today, this would not mean the Petrine office ceased at the death of Pope Pius XII in 1958. And nothing in Catholic theology says interregnums cannot be 57+ years.

^{*(from the Latin: "sede vacante" meaning "vacant chair")}

Instances of popes being involuntarily deposed by conquering sovereigns and replaced with one sympathetic to that conqueror.

A specific example is need to answer this question. See below.

Instances of contention over the papacy by two or even three popes simultaneously, accompanied by multiple colleges of cardinals and bishoprics.

Ah, the Great Schism. At the time, due to practical politics, the stituation was very confusing (I want to be loyal to this Holy Father, but the King is going in a different direction). However, in retrospect, the situation is easily understood, once the confusing cloud of Medieval politics fades.

Instances of nepotism and dynastic possession of the papacy by aristocracy and politicians.

Instances of popes engaging in orgies, conspiracies, accepting bribes, fathering children, and even committing murder.

A Bishop is per se a man who has received the fullness of Holy Orders. What his actions are are irrelevant to the definition of a Bishop. The Pope can do outrageous things, but that doesn't effect the validity of his office, for it was founded by Christ Himself (I'm sure you know the Matthew quote ;-) ). The personal Holiness of a Bishop is per accidens in regard to his Office, just as the languages he speaks and his race are "coincidences." To put it simply, Holiness of the Office does not necessarily relate to Holiness of the person in the Office.

However, those with Holy Orders do have access to special Graces, although many (as you can see) have rejected them. But look at what happens when they do accept them!

Instances of the papal office being sold.

A specific example is need to answer this question.

there is no other Church linked to any other Apostle by an unbroken chain of successors.

This is false. The Pope of Alexandria, for example, still possesses valid Holy Orders, and can trace them straight back to St. Mark. All the Eastern Churches have Succession as well. The Indian Christians were never techniquely out of communion with Rome (although they we're out of communication for 1000 years), and their Orders seem to trace back to St. Thomas.

The Bishops do not necessarily need the Pope of Rome to transmit a valid Holy Orders (otherwise he would have to be present at all Ordinations). It is considered "illegal" if done in a church outside of the Catholic communion, but it is still valid nevertheless. The Church isn't a monarchy, with the Pope as King; it's more like a federal system, with the local Bishop at the bottom, taking care of his flock, archbishops next, Patriarchs, and finally the Pope of Rome, the successor of St. Peter. The idea is not that the Pope will micromanage everything, he already has too much to do in Rome anyway. The local Bishop is suppose to work together with his priests for those people in their jurisdiction, and if a conflict outside his area arises, he can appeal to a higher authority for assistance, or the higher authority, seeing the need to intervene against the local bishop, can do so. Remember, this is the ideal; due to original sin, reality doesn't always work this way, sadly. In a sense, all bishops are ultimately equal (and powerless individually; only the College, and then, only the College in relation to the Church, is protected from error. Only the Bishops as a whole have authority), but the Roman Bishop, being the head of the College of Bishops, has certain powers necessary to resolve conflict within the College itself, otherwise the College will collapse on itself. The fact that Christ foresaw the need for all this is evidence for His Divinity, IMHO.

The Bishops are Successors of the Apostles. In fact, they are Apostles. All those powers and responsibilities that Christ gave to St. Peter, St. James, St. John, etc., like the power to "bind and loose" and the responsibility to "feed My sheep" have been passed down, handed down, to the Bishops in an unbroken Tradition. The Father sends the Son, the Son sends the Apostles, and the Apostles send their successors, and those successors send their successors, all the way down to Bishops like Pope Francis today, who will send their successors to the next generation, until the return of our Lord.

Just from a secular point of view, the fact that the Bishop of Rome's line can be traced down to the first century AD is just astounding, to say the least.

Christi pax.