How did the jurors get to be selected in 1957's 12 Angry Men? None of them seems to have law as their profession so what makes them competent to sit in a jury?
In the US legal system, as with many others around the world, a jury is selected from the general population of registered voters by random ballot.
During the initial trial phase, known as "jury selection" both legal teams (defence and prosecution) have the right to question and then remove certain members of the jury. It's common that anyone with a genuine legal background will be removed by one or other of the teams. Certain other professions are also often "struck out" such as policemen, doctors and those with a long-standing military background.
To answer your specific point about how a jury has the capability to make a judgement on fine points of law, the simple answer is that they aren't expected to do that. The jury receives very close direction from the judge as to what they're supposed to be making a decision on, something which invariably boils down to the question of whether they feel that there was sufficient evidence for a conviction and whether they feel that the witnesses were credible.