Corrected entry: Krall did not need to use the weapon to attack York Town, he could have easily have done that with the ships that he had.
Corrected entry: When Kirk and Jay-lah appear on the bike they are already travelling at speed. This is impossible, as they would have had to enter the transporter room at speed, not possible with the Franklin's transporter.
Correction: In the first movie, the transporter was used to rescue Kirk and Sulu in the middle of a long fall...They didn't splatter on the transport pad when they materialized, so it's reasonable to assume the transporter can compensate for relative velocity of the object being transported...And it also would be reasonable that an object can be beamed giving it a relative velocity as well.
Corrected entry: During the first battle with Krall's forces, Kirk orders Sulu to "Warp us out of here." The navigational deflector had just been destroyed, so they cannot go to warp speed. The deflector dish prevents tiny particles from destroying the ship at ward speeds.
Correction: This is not a plot hole but a character error. By this point, the drones had been causing damage everywhere on the ship, so damage would not have been easy to track. Plus, Kirk was under immense stress and may not have been fully conscious of the ramifications of losing the deflector. Most importantly, Kirk was trying to get away as quickly as possible. He didn't know if the drones were warp-capable, so warping away in spite of losing the deflector was a necessary risk.
Corrected entry: It is stated that the USS Franklin is the first starship to achieve warp 4, despite being commissioned after the Federation was formed. Enterprise NX-01 in the series Enterprise achieved speeds in excess of warp 5 long before this, which predates the Kelvin timeline change.
Correction: The Franklin has only cargo transporters because person transporters were introduced with the NX-01. The Franklin is pre-Enterprise. In fact it's even pre-Federation - it was reclassified as a USS after the Federation was created.
Correction: And this is an alternate reality, remember.
Corrected entry: At the end when the attacking ships have their communications interrupted by the music, surprisingly Spock and Bones' ship isn't affected.
Correction: As all of the enemy ships did not instantly explode the instant the music started, it's reasonable to assume that Spock and McCoy would be able to evade the signal (I believe that was mentioned in the movie)... It's also reasonable to assume the Spock would figure out a way to disable the connection so their ship would not be in danger.
Corrected entry: When Jay-lah and Scotty go to check her trap, Scotty picks up a large stone. You can tell it's a prop as there is no weight to it when he picks it up.
Correction: There are many types of rocks even on Earth that are very light, and they're on an alien planet.
Corrected entry: When the Franklin is in freefall from the cliff, all of the bridge crew are leaning forward, but they should be being pushed back into their seats.
Correction: Think about it...they are falling DOWNWARDS towards the ground so surely gravity would be pulling them forward, in the same way that when they were going UP gravity would be pushing them back into their seats. And yes acceleration may be a factor but the countering forces would not be enough at this point to push them back into their seats.
Chiming in even though I haven't seen the movie. Two objects will accelerate in a fall at the same rate unless acted upon by different forces (see Galileo circa 1590). At best the crew should be weightless since gravity is acting equally on them and the ship, not pushed backwards, but the ship would likely have forces pushing back against its fall, like air resistance, so the crew would be pulled forward by gravity more than the ship - more so the faster the fall since air resistance against the ship would increase. It's like your parachute falls relatively equally to you until you increase its air resistance by deploying the chute.
Correction: The ship wasn't moving very quickly as it broke free, so it's likely the crewmen were hunching forward so they wouldn't be pushed back in their seats and away from the controls... The same thing would apply as the ship fell off the cliff. The acceleration wouldn't be enough to overpower them, even in free fall.
Correction: Attacking with his fleet would have meant damaging or destroying most of the station. As stated in the film, he wanted the station intact so he could use it to attack the Federation. The weapon would have killed every living thing in the station without damaging the infrastructure.