China (Chinese: 中国; pinyin: Zhōngguó), officially the People's Republic of China (PRC), is a country in East Asia and is the world's most populous country, with a population of around 1.428 billion in 2017. Covering approximately 9,600,000 square kilometers (3,700,000 sq mi), it is the third largest country by area. Governed by the Communist Party of China, the state exercises jurisdiction over 22 provinces, five autonomous regions, four direct-controlled municipalities (Beijing, Tianjin, Shanghai, and Chongqing), and the special administrative regions of Hong Kong and Macau.
Yes, we are allowed to pray at home especially when the Mosque is closed. Friday prayer has its conditions where ever it was established e.g. minimum number of participant is five, and the Imam of Jum'ah must be 'Adil (pious believer who does not commit sins) and other conditions.
According to the view of our esteemed jurists, attending Salat al-Jumu'ah (Friday Prayer) is not wajib, but only Mustahab, and you are able to either attend, or pray your Dhuhr and 'Asr at home. You have this option to choose.
Of course, it is always recommended to pray in the form of jama'ah, or congregation, so it is encouraged to try to pray your wajib prayers in jama'ah in any opportunity you have.
With prayers for your success.
Yes, it is highly recommended to recite Adhan and Eqamah for every obligatory daily Prayer even if you are alone. The Hadeeth says that two rows of angels will pray behind you if you recite Adhan and Eqamah, and if you recite Eqamah only with out Adhan, one row of angels will pray behind you. If some leaves both Adhan and Eqamah, he will pray alone.
Obviously, there should not be any danger on you if the government does not allow loud voice of Adhan.
This story is mentioned in the book Ithbat al-Hodat, compiled by al-Hurr al-'Amili (vol. 4, p. 28).
The gist of the story is a king in China killed his daughter and a member of his court, regretted it, and sought advice from his court on how to bring them back to life. One of the viziers said there is a man in Medina named al-Hasan ibn 'Ali who could do this. The king told him he had a month to bring al-Hasan there (even though it was about a 6 month journey from China to Medina). The vizier prayed intensely, and, while he was praying, he heard a voice saying "Rise". He rose and found Imam al-Hasan (A) there. Imam al-Hasan (A) passed his hand over the two people who had been killed and prayed that Allah revive them, and they came back to life. Then the king married his daughter to Imam al-Hasan (A).
I am not aware of a chain of narration for the story and so it is not possible to say from that angle whether it is authentic or not. Also, I am not aware of any historical evidence that Imam al-Hasan (A) went to China. So, it is possible that this story is not historically true.
However, a number of hadith indicate that the Imams had the ability to miraculously travel from place to place instantaneously and perform karamat through the permission of Allah, and so that main idea in and of itself is reasonable and supported by other texts, even if this particular story might not be true.