"Being the one, is just like being in love. No one can tell you you're in love, you just know it. Through and through. Balls to bones."
The Oracle never does anything supernatural. Its predictions could as well be based on what will be the only reasonable choice and information obtained in normal/program means.
No one knows you are in love before you know it, atleast in a subconscious way, you get nervous or behave differently in your love's presence, at which point someone else could tell from the signs. So her analogy may be exactly true: Unless you know you are the one, even if only subconsciously, she can't tell if you are or not. This would explain her "tests", like "open your" and what she says afterwards and also the sentence "Who knows? That's the way these things go.".
So the reason for her saying that he has a gift may as well be, that he is the only one who was in doubt, when asked whether he was the one or not, while others rejected right away.
So now to his death. There are 2 deaths in medicine, heart dead and brain dead. If your heart dies your body's oxygen supply is cut off. After all oxygen is used up, your cells start dying, including those in your brain. This is why people can be revived after a heart death, by pressing the chest in a rhythmic way or by electric currents, but not after being shot in the heart or head.
Agent smith never shot Neo in the head. The first and all subsequent shots went into his body. That means that his brain was fully functional, for atleast 1 minute after his death. This is also the reason why he can revive Trinity.
We also don't know where exactly the life signs are read from, because they show a 3D picture of his heart with a lot of text data on the sides. So maybe the image is obtained from the matrix data and doesn't necessarily show Neo's real heart.
Now to the kiss. From the deception earlier in the movie, where the rogue mad guy killed half the crew, we know that doing something in the real world will have the same effect on people in the matrix that are hooked up. So the kiss from Trinity will have given him the feeling on his lips of being kissed, while he may not have been able to hear her words at all. This sensation may have shocked him and let him realise that he was not in fact dead and his real body was still intact, although his mind only could have been damaged, because of the mind link between real world and matrix. This realisation may have been enough to let his brain give the right signals and let his real heart pump again, which leads to his revival in the matrix (as well as in the real world). This rise from the dead may have given him the confidence boost to believe, that he was the one, which may be all that is necessary to unleash those powers.
After all the oracle says in Matrix Reloaded, that Neo has turned her into a believer and in the first part she never mentions his revival in a convincing matter in her prophecy. She only says, "Your next life, maybe.", which could really mean anything. So his revival may have been a surprise for her after all.