Could he theoretically catch the bullet (and put it in his pocket) ?
Absolutely. Let's assume he's running at the same speed of the bullet. He reaches out, grabs the bullet, puts it in his pocket. Since the bullet is going the same speed as before, the only difference is the energy put into changing its vectors, which are next to nothing.
Why would this work? For the same reason that two relay runners can hand a baton between them. The forces involved at the moment of the hand-off are minimal because everything's moving in the same direction at the same speed.
What happens when he touches the bullet, does it burn ?
It depends on how soon after leaving the muzzle of the gun it's taken. Remember that the bullet is slowing down with each passing moment, and it's cooling off, too. Grab it right out of the gun... hot. Grab it at the far end of its arc? Possibly too hot to hold comfortably, but not hot enough to burn.
It also depends on the size and nature of the bullet. For example, a .22 long-rifle bullet doesn't have a lot of bang behind it, so not as hot as the nearly .22 calibre bullet fired by an M16 combat rifle, which has considerably more bang behind it (those are two very different bullets, BTW). Likewise, the .22 shell is likely hotter than one made out of depleted uranium (given the same amount of powder behind them) due to the higher density of the depleted uranium shell. Basically, size of the bullet, shape of the bullet, material used to manufacture the bullet, amount of powder behind the bullet, etc.... it's a complicated question.
so, the best answer to this question is, "it depends."
Where does the kinetic energy of the bullet go?
So long as the runner keeps running it doesn't go anywhere. The bullet is still moving at the same speed it was before and has the same kinetic energy.
The problem is when the runner decides to slow down. Let's assume that's instantaneous. The bullet (and his shoes, eyeglasses, rings, piercings, wallet, etc.) all want to continue at the same speed as before. This is Newton's First Law, said simply, "an object in motion remains in motion until acted upon by another force."
What is that other force? If it's in his pocket, that other force is the strength of the fabric of his pants against the decelerating force of his body. In short, if he stops instantaneously, the bullet tears through his pocket and keeps going just as it would have had it hit the pants while hanging on a clothes line.
But if the deceleration is slow enough, then the kinetic energy is absorbed into the pants and body. If you throw a rock against a bag of flour and it doesn't pierce the bag... that's what would happen. The rock stops, the bag and flour absorbed the kinetic energy.
It's worth noting how important it is that your runner decelerate very slowly. If he stopped instantly, his wallet, which has greater mass than the bullet but has been acelerated to the speed of a bullet, would want to continue. It's big enough that it might not break the skin... but it would hurt something awful.
A very practical example of this is stomping on your brakes to quickly stop your car. Where does the energy of your upper body go? Into your arms and seat belt. If you didn't have them, it would go into your head as it bounces off your windscreen.