1. Thanks for providing some information concerning the Roman Catholic practice of penance.
2. I find no biblical support for the confession of one's personal sins to a priest, minister, rabbi, or whoever. What I do find is a) confession of our sins to God, and b) confession of our sins committed against another to the one against whom the sin was done. (2Sam 19:19; 2Chron 7:14; Matt 18:15-17; Lk 15:17-21; 2Cor 7:9-11; Eph 4:32; Jam 5:16)
The practice of confession has certainly come a long way. At first, confessions were done openly in the congregation and then moved (quickly!) to the privacy of the confessional. The Bible doesn't say anywhere to confess to God, but it's certainly implied. Even if you observe the passage, "confess your sins one to another" it must necessarily be concluded that accountability to the church and accountability to God are one and the same. It isn't either/or. The idea here is that when someone sins, they offend not only God, but the Church and both relationships must be restored. The most efficient way to do this is by the presbyter who represents both God and the church in absolving of sins.
The power to forgive sins was given directly to the apostles by Christ before His ascension. "Whose sins ye forgive.." implies a commissioning of direct authority to forgive sins in the name of Christ. It needs to be noted that the 12 apostles were the leadership
of the church and it's a mistake to assume that every clause of Christ's commission was given to all Christians equally. "Did I not choose you twelve?" Jesus asked. And so when Christ ascended, the leadership structure of the Church was already in place. If there is any doubt that the apostles each held an office, it ought to be satisfied with the appointment of Matthias in Acts as a replacement to Judas. If the apostles held no office and were no different than any other Christian regarding authority, then there would be no need to replace Judas.
The successors of the apostles are bishops and as we see the early church forming, each synod was governed by a bishop and that bishop appointed presbyters to represent him. So when a priest hears a confession and gives absolution, he does so by the authority of the bishop, who is a part of a long line of succession of bishops going all the way back to the Apostles...who were authorized by Christ to forgive sins. I'm sorry my answer is so longwinded, but it is necessary to lay down some groundwork to answer your question.
1. Again, I find no biblical support for the "assigning of penance... in the form or certain prayers, charity, or acts of kindness to another." However, what I do find is that true repentance includes not only the acknowledgment of, confession of, turning from sin, but also the resolve to new obedience in regard to that sin. This 'resolve' includes both the ceasing and desisting of the sin but also, when possible, the making of reparation. (Lk 19:8; Eph 4:28)
And we are really close on this one, Pilgrim. It would be great if everyone, like Zaccaeus, voluntarily offered pennance for their sins. I'm sure you understand my point of how deftly an act of righteousness can turn our wayward feet on the right path again. When a child has transgressed, parents look for a sign that the child fully understands what he did wrong and is truly sorry. The most efficient, effective, and credible sign that a child is sorry is to do something good. A good act goes way beyond words when it comes to conveying remorse.
2. Looking at Luke 19:8,9, which you offered as a proof text as a "reason for penance",
a) There was no confession to a priest (intermediary), but rather Zacchaeus spoke directly to God; Jesus Christ.
b) There was no assignment of penance by Christ but rather Zacchaeus himself expressed his conviction of sin along with the resolve to new obedience in his giving half of his wealth to the poor and his willingness to restore four-fold to those who he had cheated. These acts were the result of the inner working of the Spirit; a mark of grace.
It also should be noted that this example is used to buttress the point about the effect of reparations to get someone out of the rut of sin. As you can imagine, Zacchaeus had grown accustomed to an income greatly supplemented by dishonest dealings. He was not only sinning, but he was in a rut of sin. Saying "sorry" would have been insufficient to affect a true change of course. This was the purpose for this example from the gospels.
So we are clean before God, but our inclination to sin remains until, by an act of righteousness, we set our feet firmly back on the path of righteousness so that we do not easily fall into the same pattern of sin again. An act of penance is a pledge of good faith, a commitment to change our ways, and a sign that we are the sons of righteousness even when we occasionally sin. You don't even have to be Catholic to perform an act of penance and I recommend that all baptized Christians do so for the reasons stated.
Sorry, but I'm unclear as to what you are intending to relate here.

Are you saying that any act of alleged "righteousness" (which is what?) turns us from sin and helps us walk in all holiness? And, I'm wondering what this "pledge of good faith" means? Is it a pledge to God, man or both or perhaps neither? How do these acts of penance, i.e., prayers, accomplish any of the three things mentioned: a) a pledge of good faith, b) a commitment to change our ways, and c) a sign that we are the sons of righteousness?
A pledge of good faith goes along with the axiom, "actions speak louder than words". An endless cycle of sin, confession, forgiveness, and sin can ensue when acts of substance fail to accompany the confession. We grow deaf to someone saying "sorry" repeatedly when their actions do not indicate a genuine course correction that we associate with genuine repentence.
In Christ's most Sacred Heart.