1) Is there any Islamic healing like using certain recitals from the Qur'an to cure schizophrenia?2) Should a person who is schizophrenic but not showing the symptoms after treatment practise obligatory prayers, etc.?

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

The Prophet (Sallallahu Alaihi wa Sallam) said: "Allah has brought forth (sent down) ailment and medication, and has made a remedy for every disease; so treat yourselves and avoid Haram medicament". [Abu Dawood]
Reciting the Qur'anic verses constitutes a form of remedy. Allah Says (interpretation of meaning): {O mankind! There has come to you a good advice from your Lord (i.e. the Qur'an, ordering all that is good and forbidding all that is evil), and a healing for that (disease of ignorance, doubt, hypocrisy and differences, etc.) in your breasts, …}[10:57].
It is also proved in Sharia to use al-Ruqia with recitations of the Qur'an and supplications from the confirmed narrations of the Sunnah.
The Prophet (Sallallahu Alaihi wa Sallam) instructed one of his companion, complaining some pain, to put his hand on the painful spot of his body and say: "Bismillah, Bismillah, Bismillah" and then "I seek help with Allah and His Mighty from the harm that I am undergoing or that might befall", "Seven times". [Reported by Imam Muslim]
There are many Hadith concerning al-Ruqia. As for reciting special Qur'anic portions to treat the one who undergoes schizophrenia, we say that the whole Qur'an is a healing and there are no specified Surahs or verses. But, Surah of al-Fatiha has a special characteristic for healing as mentioned in a sound Hadith.
As for you question concerning if a person who suffers schizophrenia has to perform prayers or not, we state that one who is sane has to perform the five daily prayers; otherwise, he/she is exempted from such an obligation in case of insanity.
Allah knows best.