E.g. Konfuzius has living descendants in all of China. How many descendants of the Prophet are there and are there any branches of his family tree that lived on until today?
In fact the Hashemi dynasty in Jordan and the Alaoui Dynasty in Morocco claim to be descendants of the prophet () and before the Fatmids, Idrissids and Saadi's too. Most people who are called or referred to as Sayyid or Cherif/Sharif are more or less true descendants from Muhammad () via his daughter Fatimah and his cousin 'Ali ibn Abi Talib.
Well, I can tell you about the descendants of Prophet Muhammad (Peace Be Upon Him)'s uncle Hamza R.A. Balochs in Pakistan say that they are descendants of Hamza R.A through his daughter Umama R.A and her husband Salma Bin Abdullah R.A. They have centuries old poem which tells how they start living in Syria and support Hussain R.A against Yazeed. After the martyrdom of Hussain R.A they migrated to Iran and live there for few centuries. This where their language Balochi came into being but later due to bitter relations between them and the ruler, Balochs now divided into 44 tribes migrated under the leadership of Mir Jalal Khan to what is now known as Balochistan. This is just a poem which got no sanad of it, but most of Balochs think that they are descendants of Hamza R.A and I am one them. After all, they have a complete family tree back to Hamza R.A.
No, and they will continue until Day of Resurrection.
Surah Al-Kawthar clearly mentions about this fact:
(108:1) Indeed, We have granted you, [O Muhammad], al-Kawthar.
(108:2) So offer Prayer and sacrifice to your Lord alone.
(108:3) Verily your enemy alone has been cut off from the roots.
In the last verse the word used for cut off is
abter. Which the some of Quraish used to abuse holy Prophet (saw). One of the meaning of the word implies to having no descendants (literally having no [living] male child). So Allah (swt) revealed this surah (chapter) as an answer. Saying to them it is you who is abter. Not the Prophet (saw).
No. All descendants of Muhammed are not dead.
It was narrated that Sa'eed bin Musayyab said: "We were with Umm Salamah and we were discussing Mahdi. She said: 'I heard the Messenger of Allah (ﷺ) say: "Mahdi will be one of the descendants of Fatimah." Sunan Ibn Majah 4086, Book 36, Hadith 161
Narrated Umm Salamah, Ummul Mu'minin:
The Prophet (ﷺ) said: The Mahdi will be of my family, of the descendants of Fatimah. Abdullah ibn Ja'far said: I heard AbulMalih praising Ali ibn Nufayl and describing his good qualities. Sunan Abi Dawud 4284, Book 38, Hadith 6
Narrated AbuSa'id al-Khudri:
The Prophet (ﷺ) said: The Mahdi will be of my stock, and will have a broad forehead a prominent nose. He will fill the earth will equity and justice as it was filled with oppression and tyranny, and he will rule for seven years. Sunan Abi Dawud 4285, Book 38, Hadith 7
The above Hadiths have been classified as Hasan. As can be observed that Imam Mahdi will be from the line of Fatima (RA) who is the daughter of the Prophet Muhammed (SAW). So we know that atleast till the emergence of the Mahdi, the descendants of the Prophet will inshallah exist.