If we run out of battery and then charge iPhone, will it turn back on? I just called by my lawyers 133 times for some very important message. My iPhone is just 3 meters in front of me being charged.

Finally my lawyer come to my apartment and we both saw that the iPhone is fully charged but not turned on.

I thought charging iPhone would automatically turn it on.

I did more research. I found out that most people have the opposite problem.

Why does my iPhone turn on when I plug it in to charge?

People are complaining about the iPhone keeps turning on and  I have the opposite problem.

Actually how does iPhone officially behave? If I charge it, should it be turned on by itself?
When an iPhone is plugged in, it will boot up (that is actually a common techniques used for restarting an iPhone if the power button doesn't work for some reason). 

If the iPhone is powered off while charging, it will remain powered off and will not automatically come back on. 

I understand that you have an iPhone 4 and there is the chance that it was having trouble receiving calls unrelated to the power. However, if the phone was powered off when you checked, that is the simple answer as the cause of the missed calls.