For example say, someone is taking care of a patient and he mistakenly gave wrong treatment to his patient. If that patient dies for that reason, can we say that the patient died before her fate death came due to the mistake of the patient's attendant?
The question answers itself.
If her death is fixed due to medical fault then she would pass away due to that reason. That she would die due to medical fault is blotted for her.
Otherwise the word fixed would not bear any meaning. So, no one can die before or after.
This is a part of belief for the Muslims. A believer must believe in it that whatever that comes at his way is blotted for him; which we call fate.
وَأَنَّهُ هُوَ أَمَاتَ وَأَحْيَا
That it is He Who granteth Death and Life; [53:44]
No. At first, how do you know that the time for death was some another time in the future? Nobody can know when anybody will die, nor their future, nor their fate. We can say only the time of death, after death has occured. By definition, the time of death is when somebody died, which we can only know after death. We don't know fate, thus we can't argue that the real time of death is different from the time of death in the fate.