1) If I joined a prayer in the third Rakah of Isha prayer, I will finish my two Rakah with the imam. Then I have to stand up and recite opening supplication, Surah Fatihah and Ayah, or only Surah Fatihah. That means I will recite Surah Fatihah in all four Rakah. Please inform me about this. I will be grateful to you.
2) I read in one of the book that reciting Ayatul Kursi 313 times between Asr and Maghrib on Friday is very good for reward. Is there any Hadith supporting this.
3) My niece is suffering from cancer. Please advise me of some Dua to be read.

Praise be to Allah, the Lord of the Worlds; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

The Rak'ah of the prayer you missed with Imam will be the first part of your prayer and what you completed with Imam is the last part of your prayer; it is not making up the missed Rak’ah.

However, it is better to recite another Surah after al-Fatihah in your last Rak’ah if you cannot recite it with Imam.  Imam Ibn Hajar Al-Haitami wrote in his explanation of Masail Al-Taleem, 'If one missed reciting a Surah with al-Fatihah in his first two Rak’ah he has to recite them in last two Rak’ah otherwise his prayer will be without the additional Surah with Surah al-Fatihah. Similarly, if one misses his first Rak’ah with Imam he may recite the additional Surah in his second and third Rak’ah'.

Know that this is the opinion of Al-Shafi’eyah and we believe that it is the preponderant one.

Allah knows best.