Other than Biggs telling the commanding officer that Luke the "best bush pilot in the outer rim", what other qualifications did he have to lead the fighter group that took out the Death Star? Both Biggs and Wedge had been in the rebel alliance longer and I'd assume seasoned pilots, at least more so than Luke. Was there a reason that Biggs or Wedge didn't lead the last group? Up until that battle, I don't remember reading or seeing anything about Luke being in combat. I wouldn't think you'd want someone that green leading an assault as important as that one was.
In-universe: Red Leader had been keeping tabs on Luke during the battle. his natural talent as a pilot showed through enough for Red Leader to call for Luke to lead the second run. He certainly stepped up to it. Biggs and Wedge, as more experienced pilots who'd flown together, were logically better suited to evade the TIEs that Red Leader knew would be coming in behind them.
Out-of-universe: Luke has to take the shot. There's simply no two ways around it.
By the time the "shot heard round the Empire" occurred, Luke's entire squadron was shot down minus Biggs, who had retreated out to counter Vader's TIE squad. Luke was the only one left to take the shot. He didn't lead the squad at all, he just outlasted most of them.
In the briefing before the battle, it was Luke who was speaking up about being able to hit targets about the size of the exhaust port while flying at high speeds. While Wedge and Biggs were more experienced (and likely better) pilots, the skill at piloting does not equal the skill at gunnery. Likely a decision was made that Wedge and Biggs would use their superior piloting skills to cover Luke while he used his superior gunnery skills.
Because Luke is incorporated into the rebel army with a high starting rank in recognition of his having rescued Princess Leia. Remember that Han is also offered a commission, although he refuses.