Mutah is a marriage contract between a man and a woman where they agree on establishing a marital relationship with the stipulation of a dower, for a certain period of time. The said relation ends when the mentioned time elapses. Muslims have agreed on this legislation, but the second caliph prohibited it during his caliphate, and this became the reason for the differences in opinion on it, as history and jurisprudential books explained.

Muslims have unanimously agreed that temporary marriage was legislated by the Prophet (peace be upon him and his holy progeny). The second caliph stopped people from doing it during his rule at a famous incident. Since it was legislated by the Prophet, it remains lawful and valid. And this is what the followers of the Ahlul-bait (peace be upon him) and a group of the companions of the prophets (peace be upon him) believe.

There is no doubt that the temporary marriage is legislated. Proof is present in the Holy Book and the noble traditions of the Prophet (peace be upon him and his progeny) which are authenticated in the Sunni and Shia books. What happened after that is that the second Caliph prevented it and its prohibition then became widely accepted amongst the Sunni scholars.
What indicates its legislation from the Holy Book is the following verse: “It is a decree of Allah for you. Lawful unto you are all beyond those mentioned, so that you seek them with your wealth in honest wedlock, not debauchery. And those of whom you seek content (by marrying them), give unto them their portions as a duty." (24:4)
In the book of Nail Al-Awtar, Vol.6, Pg.270, it is mentioned that Ibn Abbas used to recite this verse and he used to add "by a set time" after "And those of whom you seek content" and this clearly explains the legislation of the temporary marriage. In this same book it is mentioned that Ibn Abbas stated that temporary marriage was a mercy that Almighty Allah has given to his slaves and if it was not the forbidding of Umar, no one would ever need to commit adultery.
In the book of Al-Muhalla, vol. 9, pg. 520, the author, Ibn Hazm, stated: "The permissibility of temporary marriage after the time of the Prophet (peace be upon him and his progeny) was affirmed by a group of his companions like Asma, the daughter of Abu Bakr, Jabir bin Abdillah, Ibn Masoud, Ibn Abbas, Moawiyah bin Abi Sufyan "The list continues on until he said: "and most jurists of the holy city of Mecca".
From this, one can conclude that the disagreement started after the prohibition of Umar and not in its legislation itself.

Verse (4:22) includes the legislation of temporary marriage based on what appears from it, since what appears from the term "seek pleasure" is that the reason behind such contract is permitted sexual enjoyment, which corresponds to the temporary contract. What supports this understanding is what Al-Tabari narrates, that Ibn Abbas and Ubay bin Ka’b used to recite this verse by adding "at a set time" after reciting "those of whom you seek pleasure…". We assume that this added phrase is their explanation to the ruling mentioned in the verse.
In the commentary book of Al-Tabari, he narrates that Mujahid and Al-Sedi said that this verse indicates the temporary marriage.
In any case, the many narrations from the Prophet and the Infallibles (peace be upon them all) are sufficient to prove the legislation of temporary marriage.

A person has a strong need for sex after puberty, whether it’s a male or a female. In some cases, it is not possible to satisfy this need through permanent marriage. Perhaps the wisdom behind the legislation of temporary marriage is to create an opportunity for permissible sex through temporary marriage so one does not fall into adultery and this can become a gateway to other ethical deviation in one's life. If a person dares to commit adultery, this leads him to commit other prohibitions.

Yes, it is permissible.

It is permissible to enter into temporary marriage for the sake of pleasure and lust via legislated means and within religious guidelines. Among the benefits of temporary marriage is that it controls sexual relation between the two sexes within specific rules and guidelines; like specifying the duration of the marriage and the dower, serving the waiting period between one marriage and another, and the child born from such a matrimony being legitimate for the couple. There is no doubt that if such marriage was common in the society, committing adultery would be minimal.

If the two parties agreed on forming a marriage contract between them in such a way that the paid money is considered as dower like the permanent marriage, then the marriage is valid. If she understands that the relationship allows her to have sexual acts in return for a fee, then it is adultery. The same can be said if she understands the relationship as a form of friendship.

It is permissible to conduct temporary marriage with a Muslim woman if she understands the relationship as marriage.
If she was divorced or widowed, then marrying her and consummating the marriage is permissible without the permission of her guardian. If she was virgin then marrying her without the permission of the guardian is permissible but consummating the marriage is not.

Temporary marriage is a legitimate way to refrain from doing adultery, but it is impermissible to use such non-religious ways to reach it. One who does not have the means to achieve it by a legitimate way, should have patience and dependence on Almighty Allah, as He said: “And whosoever puts his trust in Allah, He will suffice him” (65:3).

Yes, it applies to the temporary marriage also, unless it involves another issue that would lead to the preference of not having the temporary marriage.

If both parties intend to form a marital bond, albeit temporary, and other conditions are fulfilled, then such marriage contracts are valid.