Pike's sideburns change length throughout. In some scenes they are long enough to reach the bottom of his ear, in others they only reach about halfway down his ears.
When the crew is coming up in the elevator, you can see Spock handling and starting to put away his tricorder into his bag, but when the camera changes to a longer shot, his tricorder is away and his hand is at his side.
When the crew arrive on the planet, the security chief's tricorder screen is on the right side, later it is shown on the left side.
Christine Chapel is seen wearing commander's stripes. In Star Trek (1966) she was a provisional ensign, and later a lieutenant. She did not achieve the rank of commander until decades later.
Pike orders the Enterprise to "low orbit," but the Enterprise is then seen below the clouds. That is well below what anyone would consider "low orbit."
The officers at the two forwards stations (here identified as helm and ops) wear red uniforms, indicating security and engineering division. In the original series, the helmsman and navigator wore gold, indicating command and bridge crew.
The two forward stations ahead of the captain's chair on the Enterprise's bridge are identified as helm and ops. Those were the two forward stations on the Galaxy-class Enterprise D in Star Trek: The Next Generation (1987). On the original Enterprise, those stations were helm and navigation. Also, ops is where helm should be and helm is where navigation should be.
Chapel is able to make several crewmen resemble the native residents of Kiley 279 by injecting them with gene therapy based on the genetic code of the planet's inhabitants. As Starfleet has not made contact with the planet (other than the three missing officers from the Archer), there is no way she would have genetic material from its inhabitants to use in her work.
However, even on 21st century Earth, complete genome sequences (and other biological information) are widely available online. Chapel could easily have accessed the planet's equivalent of the internet. Moreover, Una's ship could have had samples of biological material which she could have accessed.
The bio-domes are shown in orbit around Jupiter. At that distance from the sun, there would not be enough light or heat for earth plants or animals to survive. What there is though is more than enough radiation to sterilise those domes many times over.
Seed pods would not grow into forests. The whole point of a seed vault is to preserve seeds and not to grow them into plants.
At no point were the seed pods referred to as 'vaults'. The 'pods' may have been structures, possibly the domes that Pike referred to, acting as nurseries to grow plant, as in the film Silent Running (1972).
There is no sound possible in the vacuum of space.
However, space is not a complete vacuum, just a very, very thin gas. Who knows what effect the vast energies dissipated by a ship travelling at warp or near the speed of light would be?
Spock's genetic re-sequencing wearing off somehow affected his hairstyle.
When La'an is injected with the gene therapy to make her appear as one of the native residents of Kiley 279, her eyebrows fade away as part of the transformation. They are not shown falling out, they simply fade into nothingness.
When the Enterprise officers beam down to the planet, La'an's hairstyle changes from when she dematerializes on the transporter pad to materializing in the alley, going from tightly braided to loose. Only the officers' clothing was said to be affected during transport.
The on-screen caption states that the story takes place on stardate 1739.12 but Pike's log entry says it's stardate 2259.42.
The Archer's crew manifest shows that it was manned by only Una and two other officers. Starfleet ships of that era required a much larger crew to operate. The original Enterprise had to be jury rigged up by Scottie so that a crew of just 6 could run it in Star Trek III: The Search for Spock (1984), and even then it was barely functional.
Pike is flown up to the Enterprise in a shuttle, then beamed from the shuttle to the Enterprise. Starfleet transporters are capable of beaming from a planet's surface to a ship in space. If he was just going to beam over anyway, there was no need for the shuttle ride.
Una tells Pike that the inhabitants of Kiley 279 were able to reverse engineer a matter/anti-matter device simply by observing the final battle at the end of season 2 of Discovery via telescope. There is simply no way that they could have derived any knowledge of how to construct working warp-capable devices by observing ships in flight.
Pike tells La'an that, prior to World War 3, Earth scientists sent bio-domes into space to preserve many species of plant life. World War 3 began in 2026 in the Star Trek chronology. Earth did not have the technology to send bio-domes full of life into space at that time. According to Star Trek: Picard (2020), they were still barely able to get manned rockets out.
The final shot is of the Enterprise going to warp from an orbit around Saturn. Going to warp within a solar system has always been depicted as dangerous and risky and only done in extreme emergencies. There was no emergency, and thus no reason the Enterprise shouldn't have cleared the solar system at impulse speed before jumping to warp.
After the Enterprise gets underway, Pike tells Lt. Noonien-Singh that she has the "comm." The correct term is "conn."
Pike decides to ignore General Order One because he felt the Federation had already influenced the people of Kiley 279. The people of Kiley 279's only "influence" came from observing a space battle via telescope. The Federation is not responsible for what a pre-warp civilization observes through their telescopes.
Upon beaming down to Kiley 279, Pike questions why they always arrive in an alley. The transporter officer stated they'd be set down somewhere with no foot traffic. Alleyways would be one of the few public spaces without foot traffic.