Wow, the other captains and officers on the base, whom Kirk had regarded as friends, sure assume he's guilty the instant they hear about what happened.
It could theoretically be because they're thinking that a computer doesn't lie. Also, from a Doylist position, there seems to be a general tendency in Star Trek for anyone coming from outside the Enterprise (especially people who for whatever reason Kirk can't just order to mind their own business and let his crew do their thing) to be less helpful at best and a major danger at worst.
Also, at this early point in Trek history, Kirk wasn't yet the beloved hero who had saved the Federation on so many occasions. He was still at the stage where he was considered, by many, to be a wunderkind of whom many were skeptical as to whether he was skilled or just lucky.
Also, while you say that Kirk had regarded them as friends, I'm not sure that was ever established. It was shown they were at the academy together, as Kirk was also shown to have been with the alleged dead man. However, while at the academy Kirk was apparently a pretty uptight dedicated student and not necessarily the type to have a lot of friends. So it's possible they were never Kirk's friends but were Finney's, making their reaction very understandable.
It's also a pretty far-fetched coincidence that the best defense attorney just happened to be on base as well.
It's not clear that he actually was based on the starbase. He set himself up in Kirk's quarters. He showed up after Shaw asked him to and brought a bunch of books rather than using his own library, so it doesn't necessarily follow that he was based on the starbase. He could have been based somewhere else on the planet (since the starbase was shown to be located on a planet) or even on a nearby ship. So the odds of a skilled attorney being in that range aren't actually that bad.
How, exactly, does someone tampering with a couple of data files (the logs Finney tampered with), interfere with the operation of an executable file (the chess program) that would have no reason to access said data files?
Spock mentions that he, Kirk, and Finney are about the only people on board who could've performed the computer modifications. To start with, while that means they could make the modifications, probably Spock would be the only one who could get it done without collateral damage to other programs and logs. This would make sense if what Finney did was the following:
1. After his suggested death via pod jettison, he hides from the crew and works his way to the computer core.
2. He watches the video logs of the ion storm, and works out what needs to be done to fake the logs to show Kirk cut him loose early, probably just splicing in a later pod jettison sequence in lieu of the Red Alert call and blanking out the flashing Red Alert indicator on the chair panel.
3. Presuming (for obvious reasons), there's no dedicated program in the computer to alter ship's official logs, Finney has to manipulate everything on a raw bit-by-bit level. He does this, but isn't careful in how he applies this copy-modify-paste routine, with the result that similar byte sequences in the computer get modified along with the video log, causing the chess program to go haywire.
4. Also, the records were obviously different camera angles, one from the front of the bridge, and the second was a tight focus on the captain's chair. All he had to do is tamper with the single track/angle that shows the panel, and either "photo-shop" the lights differently, or, even easier, have a track of someone wearing a captain's shirt press the buttons. Editing the two tracks together would create the illusion of Kirk pressing the button.
Analogously, Finney could've done a find-and-replace in a text file, but changed every word in the document instead of the two or three words he meant to replace. Honestly, the Enterprise was lucky to not have BSoD'd long before they got back to a Starbase.
Also why presume that computers in the 23rd century function like the ones we have? How many operating systems have there been that are rarely, if ever used anymore? There is a lot of time for many more by the time this episode would take place.
Semantics of an operating system/file system/computer architecture set aside, the Church-Turing hypothesis essentially guarantees that a computer algorithm built to run on an idealized computer (the "Turing machine") can be run on any system—past, present, or future—that supports the features of such a machine (save for infinite memory). Speed, needed storage, and other factors will vary, but in general: if you can do it on a PC in the 1990s, you could do it on a 1970s mainframe (but slower and needing more power) or a 23rd century duotronic ship's computer.
Still doesn’t mean that computers couldn't be different enough that this wouldn't matter.
It's odd how Kirk's chair has a right hand control panel with five buttons and only three visible options, those three being 'Yellow Alert', 'Red Alert', and 'Jettison Pod'; the third being not essential for command operations of the Starship. An Operations Officer could easily jettison the pod on orders from Kirk rather than it just being a button that ejects a piece of the ship. Plus there's no safety catch and/or arming mechanism on the control panel seemingly for the pod ejection system, so anyone sitting in that chair could accidentally jettison the pod if they hit the wrong button!
One could speculate that the two unlabelled buttons may serve a lock/unlock function (deactivating and reactivating the other buttons as necessary), thereby preventing accidental firing of the pod. As to why the jettison pod button would be on the Captain's chair, the only scenario we've seen where it was necessary was one where it explicitly required sacrificing someone's life. Pressing the button in that case is a pretty heavy burden to bear. It's quite possible that jettisoning a pod under normal circumstances can be done by the Operations Officer (or someone else) and would be done by giving an order. However it may be felt that if it's the Captain making the call to potentially kill someone, they should be the one to press the button (instead of forcing someone else to deal with the guilt of pressing the button to send someone to their death).
It would appear that senior cadets remain at Starfleet Academy to serve, at least temporarily, as instructors; Kirk was a student in Finney's class, and in Where No Man Has Gone Before Commander Gary Mitchell recalls that "Lt. Kirk" was something of a terror as a teacher. But we also learn that it was Ensign Kirk that made the log entry that derailed Finney's career. Did Kirk return to the Academy for a time to teach after his promotion to Lieutenant? Or did the writers just forget that his rank as a teacher had already been established?
It's almost certain that the writers forgot that his rank as a teacher had already been established. However, it's reasonable to believe that Lieutenant Kirk was training others, and so could have been Gary Mitchell's teacher without even going back to the Academy.
