Question: There were two of Khan's untraceable torpedoes in the bay that Scotty goes into but who was the one who left in the first torpedo? We never know as we see Keenster captured and Scotty's torpedo destroyed after he climbs down.
Question: When Chekov and Kirk step into their escape pods while on the Enterprise there are wearing their yellow uniforms, but when arriving on the planet they step out of the escape pods and are now wearing a completely different outfit. All of the other crew that used escape pods have not changed outfits. Where did they get the new clothes from?
Question: McCoy joins Kirk for a drink, revealing a bottle he found in Chekov's locker. Firstly, what was he doing in Chekov's locker? Secondly, what's with the two of them clinking glasses with a third drink? The locker and the clinking might suggest Chekov was dead, as they might have to fill in if Anton Yelchin had died before filming was complete, but that's not what happened. Chekov is very much alive in the next scene.
Chosen answer: Why Bones might have been going through Chekov's locker is addressed in another answer. The third glass is not for Chekov - who, as you point out, is alive - it's in memory of Jim's dad, George Kirk, who (as Bones mentions in this scene and as shown in Star Trek 2009) died on the same day Jim was born. Because of this fact, Jim's birthday always reminds him of his dad's death, so they pour a glass, and raise a toast, in his honour every year.
The director reportedly said the 3rd glass was a late addition. Perhaps it can serve both purposes, the scripted intention as well as a subtle tribute. See #7 in https://www.empireonline.com/movies/features/star-trek-beyond-justin-lin-simon-pegg/.
Question: If the characters are younger versions of the original crew, why does the picture young Mr. Spock views when he is told that ambassador Spock has died include all of the original crew members? Who are they supposed to be?
Answer: Those are the personal effects of the old/original Spock. He lived a full life with the original Enterprise crew, as documented in the TV shows and movies, then came back in time to the new/altered timeline.
Question: When the remains of the Enterprise crew on the newly spaceworthy Franklin approach the swarm of attacking fighters they broadcast a song to disrupt the fighter network. What was the song and by what musical group? (The results reminded me of the effect of yodeling in Attack of the Martians).
Question: On the way to Starbase Yorktown, McCoy joins Kirk for a drink; Kirk is already drinking something illegal. McCoy shows him something he found in Chekov's locker, and they both enjoy it tremendously. Exactly what was McCoy doing rummaging around in Chekov's locker? Was it some sort of contraband check?
Chosen answer: This scene from "Star Trek: Beyond" is an alternate-timeline version of the scene from "Star Trek II: The Wrath of Khan" in which McCoy brings Kirk an illegal bottle of Romulan Ale as a birthday gift (so we know that McCoy isn't above bending or breaking the law a little in obtaining a gift for Kirk). In "Beyond," this scene was re-imagined with Kirk already drinking illegal Saurian Brandy for his birthday, and McCoy surprises him with a very old bottle of Scotch Whisky from Chekov's locker. More than likely, McCoy knew very well that Chekov kept a prized bottle of Scotch in his locker, and McCoy simply took the bottle. What was Chekov going to do, report the Chief Medical Officer to the Captain?
Or in the future it is not above humanity to be willing to share things more than now. Maybe Chekov had given his approval for anyone to partake in a drink that he may possess, as it is no big deal.
The point is made in Beyond that McCoy "took" the scotch from Chekov's locker. If Chekov freely shared his prized bottle of scotch, McCoy should have said, "Here's a gift from Mr. Chekov, and he sends best wishes." To me, the larger question is why was Chekov hiding a bottle of liquor in a "locker" when he could have more easily kept it in his personal quarters? Perhaps because the contents of personal quarters were easily monitored, so Chekov had to HIDE it in a less conspicuous area.
Question: Why does Krall believe that the Federation abandoned him and his crew? The Federation wouldn't know where to look when The Franklin was declared MIA.
Answer: To start, Edison (who became Krall) was part of maco (Military Assault Command Operations). When the Federation was created, maco was disbanded and Edison became a Starfleet officer. When he was put in command of the Franklin, he had already begun to resent the Federation for being "put out to pasture" and felt obsolete as a soldier in an organization that promoted diplomacy, peace, and tolerance. When the Franklin crashed, he sent out a distress signal that was suppose to tell the Federation where they were at. He had no way of knowing they didn't get his distress call. Already resenting the Federation, Edison's anger grew and he simply felt they abandoned him and saw it as the Federation trying to get rid of former soldiers.
Question: Why does Spock use his left hand for the "live long and prosper" greeting to the older Vulcans?
Chosen answer: Presumably, it must have been much easier for him to use the left hand to do the Vulcan salute over his right hand. Though most people in Star Trek do the Vulcan salute with their right hand, there is no rule saying which hand someone has to use in order to do the Vulcan salute with.
Question: Why does Edison become Krall? In other words: why does his appearance change? If the tech that keeps him alive changes him, how can he change (most of the way) back? Finally, if he could change back/remain human this whole time, why would he look like he did?
Answer: The energy transference technology Krall used to drain the life of others caused him to gradually begin to look like the species he was draining. After the attack on the Enterprise he drained several human officers and began to look like his former self.
Chosen answer: The outfits were in the escape pods they escaped in. The other crew members didn't change outfits because they were all captured and were unable to. Chekov and Kirk were not captured and were also the only crew on board who successfully escaped in the designated escape pods, so logically they would be the only people wearing different outfits.