All perfect praise be to Allaah, The Lord of the worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
Those who pretend that they are the descendants of the grandsons of the Prophet, sallallaahu ‘alayhi wa sallam, who are Al-Hasan and Al-Husayn are of two groups:
The first group:
This group comprises those who have authentic sources that confirm that they are really his descendants, or that the information is spread among the people that they are his descendants and that it is impossible for this information to be false. In this case, if one denies it, he is wronging them and this is considered among the acts of Jaahiliyyah (pre-Islamic Period of Ignorance). The Prophet, sallallaahu ‘alayhi wa sallam, said: “Slandering lineages is a practice of Jaahiliyyah (pre-Islamic Period of Ignorance).” [Al-Bukhari] Ibn Hajar explained this Hadeeth to mean “the defamation of some people's lineage by some other people without any knowledge.”
They are called “Saadah” (plural of Sayyid, which means “Master”) because their ancestry goes back to the master and best of all creation, who is Muhammad, sallallaahu ‘alayhi wa sallam. The Prophet, sallallaahu ‘alayhi wa sallam, also said about Al-Hasan : "My grandson Hasan is a Sayyid –Master.”
The second group:
These are those who have no authentic sources about their descent and who are not well-known among the people to be the descendants of the Prophet, sallallaahu ‘alayhi wa sallam. Therefore, it is not correct to say that they are his descendants, because in principle they are not.
Therefore, the lineage of a person, leading up to the Prophet, sallallaahu ‘alayhi wa sallam, cannot be confirmed or refuted except by investigating the matter, so that one does not commit an act contrary to the Sharee’ah (Islamic law).
As for your second question, please read Fatwa 86095.
Allaah Knows best.