Sayyid

If your father is a Sayyed, you will be a Sayyed as well. Your mother alone with out your father will not make you a Sayyed.

Wassalam.

Being a descendant of a person or a family depends on the definition of descendant. In general you are a descendant of all your grandfathers and grand mothers. But if you ask about being a Sayyed, then the definition of a Sayyed is the person who was born from a Sayyed father. So, your grandfather is related to the Prophet's family from his mother, but since his father was not a Sayyed, he is not a Sayyed. This is the verdict of most of our great scholars.

Wassalam.

There is a common understanding among people of our Shia community that Black and Green turban and green shawl are worn by Sayyeds only.

Non Sayyed wearing a green shawl in public can make others think that he is a Sayyed which is not right. You can wear green shawl at home when there is no possibility of any one to think that you are a Sayyed.

Wassalam.

If your employer has allowed you to keep the tips, you are allowed then to keep it. It is not a Sadaqa.

Wassalam.

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.