Sayyid

Islam does not allow the needy Sayeds to receive Zakat from people, and instead of that, kept for needy pious Sayeds a share in Khums. It is a matter of keeping the respect of the family of the Prophet (SAWA) with out any differentiation or discrimination. Non Sayed needy persons are allowed to receive Zakat while needy Sayeds are not allowed to receive Zakat. Justice requires providing a source away from Zakat to help needy Sayeds which is in fact Sahm e Sadat.

Wassalam.

Sayyed families usually have family tree proving their lineage. If they don't have a family tree but known in their native place as Sayyeds,b that also is considered as a proof.

Studying in Hawza is open for every one who is interested in Islamic knowledge and does not need any proof of lineage.

Wassalam.

Yes. Aqeeqa meat is allowed from any one for any one. Parents of the child are advised not to eat from the Aqeeqa of their child. It is not Haraam for them to eat, but just Makrouh.

Wassalam.

The most learned Mujahid among Shia Ulama can be known through valid witnesses of at least two of the high caliber Ulama in the Hawza who are called Ahl al-Khibra or Khubra أهل الخبرة.

It is not for any one else out of the high caliber Ulama to give witness in this matter. 
Hawza Ulama usually guide people when the need arises.

There are few high learned Mujtahids but you need two high caliber Ulama to give you their wtnisses.

Wassalam.

1. Purity of bloodline because of the connection to Lady Fatima (AS) remains with every one who has this connection whether from his Sayyed father or Sayyedah mother.

2. There is another jurisprudence rule to define the Sayyed person who will be allowed to give his obligatory Zakat or Fitra to needy Sayyed. Most of our Maraaje' of Taqleed say that the definition of Sayyed is the person who was born from Sayyed father.
These are two different issues.

Wassalam.

Bani Hashim (the family of the Prophet) are entitled to Sahm Sadat if they are poor. The grand children of Ameerul Mo'mineen Ali (AS) are called Alawi Sadat while grandchildren of Aqeel ibn aabi Talib are called Aqeeli Sadat and grandchildren of Jafar ibn Abi Talib are called Jafari Sadat.

Wassalam.

When your Marja' has not issued a Fatwa (verdict) on a specific matter, you can refer to the second Marja' who is next highest in the degree of knowledge according to most senior Ulama of the Hawza.

Wassalam.

It is absolutely fair as all the rules of Islam are fair with out fail. Those who are linked with the greatest human being, Prophet Muhammad (SAWA) are gifted more reward when they do good and implement his teachings , and this gift is for the sake of respect and love of the Prophet Muhammad (SAWA). Similarly if they do wrong and violate the message of Islam, they will entitled to double chastisement because of the disrespect they showed to the Prophet (SAWA).

In Quran said : O wives of the Prophet; any one from you commits a proven sin will be punished double, and any one from you submits to the will of Allah and His messenger and does good, we shall grant her double of reward. (33:30 and 31).

Wassalam.