There is nothing wrong with praying for relief from harm and there is no contradiction between offering du’a for healing and being patient.

The Du`a of the fasting person that is answered is that which he offers whilst he is fasting, until he starts to break his fast, and not after that. As for the Dhikr, ‘Dhahaba Al-Dhama’u wabb talat Al-`uruq wa thabata Al-ajr in sha Allah’, it can only be said after breaking the fast.

This du‘a ‘Ya Hayyu Ya Qayyum…’ was narrated in a sahih hadith from Anas ibn Malik. Imam Al-Mundhiri and Shaykh al-Albani said: its isnad is hasan. For more about the meaning of this du’a, please see the detailed answer.

It is not permissible to single out certain verses of the Quran to recite for specific purposes, unless there is specific Shar`i evidence to that effect. Therefore, there is no specific surah to recite when in difficulty.