What is the meaning of khuffayn (leather socks) and jawarab (socks)?
All praise is due to Allah, and I invoke peace and blessings upon our Prophet Muhammad, and upon all of his Companions.

The meaning of the khuffayn is that which is worn upon the foot being made of leather and it's likes. The meaning of jawarab is that which is worn on the foot being made of cotton and it's likes, and it is what is referred to commonly as shurrab (socks, stockings).

What is the ruling of wiping on the khuffayn and jawarab, and what is the evidence for that from the Book and Sunnah?
The wiping is a sunnah which has come to us from the Messenger sallallahu 'alayhi wa sallam, and it is better for the one wearing them to wipe over them, rather than taking them off to wash his feet.

The evidence for this is the hadith of Al-Mughirah bin Shu'bah radiallahu 'anhu who said that the Prophet sallallahu 'alayhi wa sallam made wudhu. Al-Mughirah said, "I moved to remove his khuffayn and he said,'Leave them, for indeed my feet were in a state of purity when I wore them.' So he wiped over them."

The legislation of wiping over the khuffayn is established in the Book of Allah and the Sunnah of Messenger of Allah sallallahu 'alayhi wa sallam. As for it's legislation in the Book, then it lies in the verse, "O you who believe, when you stand for prayer, wash you faces and hands up to the elbows, and wipe your heads and feet up to the ankles."

There are two correct ways of reciting the saying of Allah (wa arjulikum) 'and your feet' having reached us from the Messenger of Allah sallallahu 'alayhi wa sallam:

Reciting it as 'wa arjulakum' with a fatha on the lam, hence linking it in meaning to 'wa wujuhakum' ('and your faces'), i.e. wash the feet.
Reciting it as 'wa arjulikum' with a kasra on the lam, hence linking it in meaning to 'bi ru'usikum' ('your heads'), i.e. wipe the feet.
And that which explains when one should wipe the feet and when one should wash them is the sunnah, for the Messenger sallallahu 'alayhi wa sallam used to wash his feet when they were uncovered, and wipe over them when they were covered by khuffayn.

As for it's legislation in the sunnah, the sunnah is mutawatir from the Messenger of Allah sallallahu 'alayhi wa sallam in this sunnah. Imam Ahmad said, "There is no aversion in my heart to wiping. There are forty ahadeeth with regards to it from the Messenger of Allah sallallahu 'alayhi wa sallam and his Companions", and the saying of the poet:

"From those that are mutawatir is the hadith 'the one who lies ... ',
and 'the one who builds a house for the sake of Allah, and hopes for the reward',
and the Sight, the Intercession and the Pond,
and the wiping over the socks, and ... "

So this is the evidence from the Book of Allah, and the sunnah of His Messenger sallallahu 'alayhi wa sallam.

What are the authentic, established conditions for wiping over the socks and what are their evidences?
There are four conditions:

That he be in a state of purity when he wore them. And the evidence for this is his sallallahu 'alayhi wa sallam saying to Mughirah, "Leave them, for indeed my feet were in a state of purity when I wore them."
That the khuffayn or jawarab be pure and clean. If they have filth on them then it is not permissible to wipe over them. And the evidence for this is that the Messenger of Allah sallallahu 'alayhi wa sallam prayed with his Companions one day in shoes, and during the prayer he removed them. Then afterwards he informed his Companions that this was because Jibril had informed him that they had filth or dirt on them. And this indicates that the prayer is not acceptable when there is something in it which is impure, and because when the wiper wipes over the filth it sullies him also, and then he is also not in a state of purity.
That he wipes over them when he is in a state of minor impurity, not when he is junub or in any state that necessitates the ghusl. And the evidence for that is the hadith of Safwan ibn Asal radiallahu 'anhu who said, "The Messenger of Allah sallallahu 'alayhi wa sallam ordered us not to remove our socks for three days and nights when we were on a journey except if we were junub, but not from excretion, urination or sleep." So the condition is set that the wiping is only for minor impurity, not for the major impurity due to this hadith.
That the wiping be for the period determined by the Shari'ah, and that is a day and a night for one who is resident, and three days and nights for the one on a journey. This due to the hadeeth of 'Ali radiallahu 'anhu who said, "the Messenger sallallahu 'alayhi wa sallam set a limit of one day and night for the resident, and three days and nights for the traveler," meaning for wiping over the khuffayn. [Related by Muslim] And this period starts the first time one wipes over the socks, and it finished 24 hours after for the resident and 72 hours after for the traveler.
So if we were to assume that one purified himself for fajr on Wednesday at 5am, and he remained in this state till the isha' prayer, then he slept and woke for fajr at 5am on Thursday then he has to make wudhu with washing his feet as the period has expired. But were he to make wudhu before 5am then his wiping is valid and he can pray fajr with this wiping. And he can keep praying as much as he wishes afterwards as well with that same wudhu, for the wudhu does not become void when the period of wiping ends according to the strongest opinion of the scholars. And this is because the Messenger sallallahu 'alayhi wa sallam did not set a time period for the state of purity, but he set a time period for the wiping only.

So when the period for wiping has finished then there is no wiping after this, but if he was in a state of purity when the period expired, then his state of purity remains. This is because this state of purity has been established according to the dictates of the shari'ah, so then it cannot be removed except by an evidence from the shari'ah, and there is no evidence for the state of wudhu ending as the time period for the wiping ends. And because the basic principle is that everything endures according to what it is upon until it's termination is made clear.

So these are the conditions that are set for wiping over the khuffayn. And there are other conditions that some scholars have mentioned but some of these are problematic.

A person makes tayammum, then he wears the khuffayn, then he finds water, is it permissible for him to wipe over the socks knowing that he was in a state of purity when he wore them?
It is not permissible for him to wipe over the socks when the state of purity is the purity of tayammum due to the saying of the Prophet sallallahu 'alayhi wa sallam, " ... for indeed my feet were in a state of purity when I wore them." And the purity of tayammum has no connection to the feet, rather it is only concerned with the hands and face. So building upon this, if a person does not have water with him, or he is sick and unable to use water for wudhu, then he can wear the khuffayn even without being in a state of purity, and he can wear them for an unlimited time (making tayammum) until he finds water or is cured from his illness, because the feet are not connected to the tayammum.

Is the intention obligatory, meaning that when a person intends to wear socks or shoes should he have the intention that he will wipe over them, and likewise should he have the intention that he is going to wipe over them as a resident or a traveler, or is the intention no obligatory?
The intention is not obligatory because the ruling of wiping is concerned only when the need arises and therefore it is not requiring an intention. Just like if a person were to wear a garment, it is not a condition that he has to have the intention that this garment covers his awrah when he prays. So it is not a condition that he has to have the intention that he will wipe over the socks when he wears them. Likewise he does not have to have an intention as to how long he intends to wipe for, rather if he is traveling then he has a period of three days whether he had the intention for this or not, and if he is resident he has a period of one day whether he had the intention for this or not.

What is the journey in which the wiping period of three days and nights becomes permissible?
The journey in which it becomes permissible to shorten ones prayer, is the journey in which it becomes permissible to wipe over the socks for three days and nights, this is because in the hadith of Safwan ibn Asal which we have mentioned he says, "when we were on a journey", so for as long as a person is on a journey in which he shortens his prayer he can wipe over his socks for the period of three days.

When the traveler comes to the end of his journey, or a resident starts a journey and he has already started wiping over his socks, how does he calculate the length of his wiping?
When a resident wipes over his socks and then starts a journey, then he should complete the wiping of a traveler (i.e. 3 days and nights) according to the strongest opinion. And if he was traveling and then completes his journey, then he should complete the wiping of a resident. Some of the People of Knowledge have mentioned that if one were to wipe as a resident then start a journey he should complete the period of wiping as if he were a resident, but the strongest opinion is what we have first mentioned. This is because this person has some time remaining in the period of his wiping before he started on a journey, then he started traveling so it is true to say that he is a traveler and therefore from those that can wipe over there socks for three days.