What is the biological basis for a human's ability to think clearly and be aware while "fighting/competing"?
thinking and decision making under stress or pressure is significantly reduced.  its sometimes called the 'fight or flight' condition. 

This is a pretty well established phenomenon.  Here's a great radiolab segment on stress response.  Psychologist John Gottman notes how our ability to have a conversation where we can even listen to what someone else has to say is practically nil when we are emotionally saturated (frustrated or upset that is).  Even a mildly stressful conversation with your own spouse can shut down most rational processes.  Gottman can predict couples who overcome this problem which is highly corellated with divorce btw - but most of these couples remove stress in conversations rather than learn to work under pressure as it were. 

Jonah Lehrer wrote about how tests to rank quarterbacks in their ability to respond on the gridiron are basically non correlated.  I know there have been criticisms of Lehrer, but this is a good piece and well grounded.  The point is that its hard to tell a-priori whether someone will perform under stress and given different sorts of stress.  

In the japanese space program they would screen candidates by making them do highly repetitive tasks like folding 1000 origami cranes.  NASA was just doing interviews with panels of psychologists.  That wasn't infallible. 

Such a test would be valuable for fighter pilots, astronauts, maybe presidents too! but it doesn't seem to really exist at this point. 

In the end you can winnow out those who break into a sweat in the interview, but to know who is going to really take the beach, I think that the only real answer is to see if they make it through a 'live fire' (or whatever) experience.  

Or as Clint Eastwood summed it up: "A man's got to know his limitations." 

(apologies to the ladies there - you can know your limitations too).