What's the reason the boy looking like Beckham was killed?
The boy looking like Beckham was killed because it was a manner in which the terrorists planted bombs inside dead bodies so that when people came to collect the bodies, they would detonate - thus killing even more people.
The boy was a random boy and was not killed because he looked like Beckham. So there is no linkage there.
Who killed the boy looking like Beckham?
Why was he covered with an unexploded bomb?
See answer to your first question above.
If you have seen the film No Man's Land, a similar technique is used which also becomes the focal plot point of that film.
Why James didn't believe Beckham was alive?
I believe that James hadn't seen Beckham for quite sometime and the dead boy looked similar to Beckham. Added to this, being in a war zone can be quite stressful and hence it is only normal that your mind makes you be on an edge and really alert to your surroundings. Hence even a small suspicion is blown out of proportion. (Guilty unless proven innocent in this case)