A couple of 15 year olds have summited Everest, so I guess 13 isn't that much younger. But you look at all of the young climbers who have actually summited and they are all sherpa. They have a huge amount to gain employment-wise through doing this and are much more accustomed to the altitude. Their making it to the top gives them an opportunity for a life they wouldn't be able to live otherwise.
This kid will get a lot of recognition if he makes it but there is no real need. He will live a perfectly comfortable life without this. I also feel like he would be very high risk. Not only is he putting himself into danger (which any climber is) he is putting those with him in serious danger. People are far more likely to endanger themselves for a little kid.
Is he physically capable? With sherpas ferrying all of his gear probably. If not I'm sure someone could drag him.
Should he? No