1) Why Mutaa is permissable in our fiqh its such an shameful act I guess?
Any Imam had done any mutaa? From where did this started? I dont think so its legit this is just to fullfill ones desire in unlawful way.
The question is why is it permissable when we have nikah infront of us? And from when this thing got started?

How we can assume any act shameful without knowing that correct logic and philosophy behind it. The act of Mut’a is not an act of someone who has just commanded out of his desire but we have many traditions regarding this act in Islamic traditions. But unfortunately the act of is introduced in our culture as an act of lust and because of few people as they misused this very laws for their lust nature and unlimited sexual desire normally we think that Mut’a is like an act of prostitution. While in Mut’a and Permanent marriage is that Mut’a has fixed time period while permanent marriage is for forever( unless a woman has not been divorced).

If something is considered permissible and its impropriety has been proven by the Qur'an or Sunnah, there is no need for approval of that act has been done by an infallible Imam or not, because the meaning of impropriety is that the obligee is free to do or leave it.
As Muslim Ibn Hajjaj has narrated in the correct narration that Ata’a said: Jabir Ibn Abdullah had returned from Umrah Hajj, we came to see him. People asked him about several things, including mut'a. He said: Yes, we have performed mut'ah in the time of the Prophet and Abu Bakr and Umar. The appearance of these narrations indicates that the companions performed mut'ah.
Of course, there are narrations about the practice of mut'ah by the Imams (as), such as the narration that Imam Ali (as) had a temporary marriage with a woman from the tribe of Bani Nahshl in Kufa.

‎قَالَ وَ رَوَى ابْنُ
‎بَابَوَیْهِ بِإِسْنَادِهِ أَنَّ عَلِیّاً ع نَکَحَ امْرَأَةً بِالْکُوفَةِ مِنْ بَنِی نَهْشَلٍ مُتْعَةً.
وسائل الشیعه، ج 21، ص 10.

Why mutaa when theres nikah?

Now if a person is not in position to get marry permanently and the other side a woman is also either widow or divorced ( in many case doesn’t want economical help) but for sure she want to fulfill her sexual desire through Halal way. Where should this person go who can’t marry now permanently due to any genuine reason and that woman need someone who can satisfy her sexually while she is wealthy. Though should they go to prostitution or do adultery?
Therefore Mu’ta is only for this who can’t afford permanent marriage and for those who need to be fulfilled their sexual desire without any big arrangement or something like that. Or for example a person is outside form his town for months etc.
But in short Islam say if a person can control his desires and sure will not fall into the SIn then Ok he/she can wait for there permanent marriage.
Also Mut’a does not mean a person go and find a Virgin girl and have fun just because he has permission from Shariayt no it’s not the case, Mut’a rules should not be used for Lust and misused.
It has many conditions.
So itself Mut’a is not bad but very helpful in exceptional case same as Divorce is not itself bad and very helpful for those couples who can’t live together s there life is miserable and hell better they should get separated. That is why now other religions have also accepted Divorced systems while if you go in deep neither Hinduism have the concept of Divorce nor Christianity nor Buddhism etc…,