I watched the video, and agree, no way is it fake. She is sitting on the ground, shortly after being shot in the chest. Two men, one presumably the doctor, use their hands to suppress the bleeding from her chest. She then lays backwards to the ground, and shortly after that the video clearly shows blood trickling from her mouth, then her nostrils. As you say, no effort to clear the blood to breathe.
On Fox News, the reporter commented that she was shot in the head (presumably after seeing the picture with blood on Neda's face). Fox is incorrect - she was shot in the chest, with the internal bleeding going up her airways.
There is no blood until you see a man’s hand come near her and he looks like he is pouring something on her. Then you see blood.
I can see why people think this is fake. I tend to believe it is real. I just can’t explain what that guy is doing before you see blood.
The photo they are releasing of her is not of a 16 year old unless they age pretty fast in Iran - which might be the case.
I don't see how the failure to have normal involuntary responses like blinking or coughing can be faked.