Spock is shown on the Enterprise bridge. After Pike bluffs the Shepherds into tractoring them away, Spock is shown inside a shuttle, which is outside the Enterprise on the surface of a comet. The scene on the bridge played out in real time. There was no opportunity for Spock to have exited the ship as shown, nor time to do so.
The landing party is beamed back to the ship, which then goes to red alert and is fired upon by the Shepherds. Spock and La'an, who were just seen in the transporter room in full atmospheric suits, are suddenly back on the bridge at their normal stations in their normal attire seconds later.
Pike refers to La'an's station along the port side wall of the bridge as "ops." Before, the forward starboard station was identified as ops. La'an's station was designated as tactical, which would be consistent with her position as chief of security.
The Enterprise positions itself near to the comet in order to bluff the Shepherds into not attacking them. After discussing their situation with the Shepherds, the viewscreen shows the comet at a good distance away from them.
(Around 11 minutes) Pike raises his hand to his mouth before saying 'I love this job.' He then lowers his hand in the side view shot & does so again in the next shot from the front.
Spock revives Lt. Kirk using a defibrillator function in his space suit after Kirk's heart has been stopped. Defibrillators do not restart a stopped heart. Their function is actually to stop a heart so that it may naturally restart in a normal rhythm if a patient is experiencing dysrhythmia as the result of an event such as a heart attack.
The rain on Persephone III makes no impact on the surface, nor on the inhabitants except for one closeup of two peoples' hands. The rain was a special effect added in post-production, likely due to the difficulty of generating faux rain in a desert location shoot and the difficulty in waterproofing full alien makeup.
Before beaming to the comet, La'an warns her team that the gravity there is a fraction of Earth-normal. Once there, they walk around as though under typical Earth gravity. If gravity was a fraction of Earth's, that would be impossible. That's why footage of astronauts on the moon show them bouncing and leaping rather than walking.
After the comet's shields are lowered, as the away team is beamed aboard, the transporter room door open and Chapel and her team rush in to assist. It would have made more sense (although it would have had less dramatic impact) for them to have been on standby in the room already.
Pike tells the Shepherds that the trilithium resin in the Enterprise's warp core will detonate on impact with the comet and destroy it. Trilithium was considered an experimental substance in the 2370s. This story takes place in the 2250s. Additionally, trilithium is a nuclear inhibitor that stops fusion reactions, which would be antithetical to use as part of a ship's propulsion system.
The crew of the Enterprise somehow missed the Shepherds' ship approaching and firing on them. Even if the ship had been cloaked (which there is no indication it was), the Enterprise's sensors would have picked up incoming weapons fire before they were hit.
Spock's shuttle flies into the comet, adjusts its trajectory, and flies out again without being spotted by the Shepherds. The Shepherds were said to have more advanced technology than Starfleet. There is no way they wouldn't have spotted the shuttle returning to the Enterprise at the very least.
The comet is stated to impact the planet in two days' time. After the away team travels to the comet and back, a journey of a few hours at most, they are said to be down to one hour from impact.
General Order One (aka the Prime Directive) states that Starfleet is forbidden to interfere in the development of pre-warp civilizations. Intent does not matter. Attempting to change the comet's course to avoid striking the planet is an act of interference. While GO1 is often bent or ignored all together, no one on the Enterprise even mentions it when discussing what to do about the comet.
In Uhura's opening log entry she refers to an "ancient comet". All comets are ancient, as they are left over from the creation of the planetary system they orbit. Uhura would know this.