How is John a Master Chief Petty Officer - the highest enlisted rank in the Navy - already? The Spartans are all still teenagers, so they'd be junior enlisted, or according to the books, Petty Officers.
John being a MASTER Chief Petty Officer is a continuity error - He was only Chief Petty Officer at the time of this series according to expanded universe material.
It's not stated that he's a Master Chief Petty Officer. HE just says that he wants the cadets to call him that. He may just be trying to sound badass (inasmuch as a generally unsociable super soldier would want to), or in a sense making a self-fulfilling prophecy regarding his ultimate promotion.
Or this could be a film made in-universe years and years after the conflict, based on memoirs of the survivors of the squad. It would explain the inconsistencies- memories can dull and fade, and certain liberties could be taken in order to make sure the "Audience," in the UNSC would know who was who.
Considering the opening suggests that Lasky is the one remembering the events, it could be that he is simply misremembering the events slightly, such as John's rank and Kelly and Fred's armor resembling later MJOLNIR models.
There's a reason why John-117 asks Lasky if he can operate a Warthog. Since he's only fifteen at the time, he's too young to drive.
While kinda an amusing thought, considering he was conscripted at age six and heavily trained to operate all forms of military hardware, he's probably certified to drive a Warthog. He probably felt that it would be best if he manned the turret since he had the most armor and combat experience.
Similarly, some fans complained that his voice was off and he didn't sound deep enough. Given his age, it shouldn't be.
Some people also complained about Chief being a little too openly compassionate. Again, he's far younger (and probably somewhat less jaded) than the Chief we see in the games. Also, we have to consider that this series takes place only about a year after he was forced to abandon his best friend Samuel-034 to die; he's probably still somewhat reeling from the loss.
When Vickers makes the comment "It's a shame, Mommy Laskey's only son isn't coming home", it causes Tom to go ballistic. Why? His brother is already dead.
Just before the space elevator falls, you see 3 Covenant Corvettes come into the atmosphere. This may imply that all the UNSC ships in orbit are destroyed, unless the battle in space is so heated that some managed to get through.
Not necessarily. The Covenant bypassing orbital defenses to attack ground targets is a fairly standard tactic for them. Either that or some part of their forces engage the UNSC while others proceed straight to the planet. Given that the Spartans and the surviving cadets are able to evacuate by Pelican at the end, it's likely that the Covenant fleet has still not been able to completely destroy the UNSC space presence, or was eventually defeated by UNSC reinforcements, or just left after having eradicated the human population of the planet.