The first episode invokes Horseback Heroism by showing Pike riding a horse across the Montana countryside, but this is actually an inversion of the trope, as Pike can't be a horseback hero on Earth like the cowboys of old and also be The Captain of a starship exploring the frontier. Earth is the very cradle of humanity, and as far from the dangers of the galaxy as one could hope to get.
He's also galloping away from the shuttle, metaphorically fleeing from his duty — and failing, as the shuttle and his duty overtake him.
The main cargo bay is located directly beneath the shuttle bay, even having a large hangar door opening to space. This likely makes it easier for shuttles to load and unload cargo. There is probably even an elevator for moving shuttles and cargo between the two bays, similar to the flight deck and hangar deck arrangement on aircraft carriers.
In Memento Mori the crew all wear badges dedicated to ships that they lost friends and loved ones on. Captain Pike wears a badge for USS Discovery, which was officially lost with all hands at the Xahea system. The badge can be both metaphorically true for Pike, as the surviving crew is effectively lost to him due to time travel, and also literally true, as it commemorates individuals lost as well as the ships they traveled on, and Pike lost several officers while commanding Discovery, to include Ariam and Connolly.
"A Quality of Mercy" explains why Spock was so hell-bent to get the wheelchair-bound Pike to Talos IV in "The Menagerie" to the point of disobeying Starfleet regulations and risking court-martial — he learned that had Pike not wound up in that chair, he would've suffered that fate. This isn't just mere loyalty to his former captain; it's his way of repaying a debt.
The same episode also shows why Kirk and Spock were such an effective team; the two of them were both lateral thinkers who weren't afraid to play Pragmatic Hero when the occasion called for it. Since Pike remained Captain of the Enterprise in this alternate timeline, Kirk and Spock never had a chance to develop that rapport, which is exactly what that specific conflict with the Romulans needed. Pike tried to play it by the book, but "Balance of Terror" required someone to think outside the box.
Kirk is not happy with Pike's strategy, but is willing to play ball with him. Even after the loss of the Farragut, he continues to be a team player after initially giving the other captain a What the Hell, Hero?. So why does he keep helping Pike even though Kirk has not been above trying to get around or get the better of antagonistic captains in the original timeline? Well, because Pike is neither insane nor cruel. He is failing because he won't abandon Starfleet's ideals, and Kirk respects that enough to keep backing his plays and doing what he can to help Pike succeed. This is in contrast to the relationship between the Romulan commander and his first officer, who undercuts his commander's authority at every turn because the commander is not living up to the ideals of the hawkish Romulan Star Empire.
During a pitched space battle, Spock dies as a result of injuries received repairing critical damage to the ship, including severe radiation burns, a fate which his captain is willing to do anything to reverse. "A Quality of Mercy" or "The Wrath of Khan"? It seems where Pike is destined to his fate to save his cadets, Spock is destined to his fate to save the Enterprise. And just as Pike will have a new life (figuratively) thanks to Spock bringing him to the Talosians, Spock will have a new life (literally) thanks to being deposited on Genesis and later rescued by his friends after his resurrection.
Pike's failed attempt at a diplomatic resolution against a Romulan scout attempting to instigate a war is very similar to the situation faced by Georgiou at the Binary Stars, down to their second in command being a hawkish officer from the House of Sarek advocating for a first strike.
At the end of the pilot, Pike is chastised for breaking the Prime Directive by intervening directly in the Planet of the Week's internal politics, and it's mentioned that Starfleet is going to attempt to re-brand it to get starship captains to be less cavalier about it. This suggests that the extreme Obstructive Code of Conduct treatment of the law in the TNG era may have been a result of this effort having Gone Horribly Right: changing the culture of an entire organization this way can take decades.
It’s a good thing no one was hurt in that sword fight in "The Elysian Kingdom". Ortegas as Sir Adya seemed awfully eager to kill people with that sword.
"All Those That Wander" establishes that the Gorn reproduce by implanting parasitic eggs into humanoids. How could a species evolve like that... unless there was a second sapient species native to their homeworld, whom the Gorn now use for breeding stock?
There's also the problem of what will eventually happen to everything Uhura learned from Hemmer not long after Kirk's Enterprise receives a distress signal from the Malurians.
Given that ordinarily the Klingons would be all too happy to see the Federation and the Romulans tear each other to shreds for decades, one can only imagine how truly awful things must have gone for the Alpha and Beta Quadrants as a result of the Federation-Romulan War in the Bad Future of "A Quality of Mercy" for the Monks of Boreth to be so gung-ho about preventing Pike from altering the Prime Timeline.
It is worth noting that the Klingons have, at various points in the Prime Timeline, sided with either the Federation or the Romulans. It's possible that the Federation is still in the fight 20 years later in part because the Klingons threw in their lot with them, and then things got so desperate as to cause the monks to break their time travel taboo.
It’s also been shown that while the Klingons view the Federation as weaklings too willing to accept dishonor in the name of maintaining peace and consensus (rather than defending their own view of what’s right through violence), they also see the Romulans as chronic backstabbers who would sooner dishonor themselves than engage in a fair fight. When the chips are down, at least the Federation plays fair.
