I wrote all this before I understood that the desired focus was on video games specifically:
I don't know a lot about story writing for video games or coding, but I do have experience with collaborative writing and I expect writing for video games is similar to other collaborative projects, like movies, TV shows, musical theater, etc.
The first division of labor happens at a macroscopic level and the separations are pretty clear. Usually writers are literally writing, and their products are things written on the page. In the case of popular musicians, recordings of ideas and demos of entire songs may be used instead of sheet music, but for music composers in almost every other genre, their product is written down or put into a computer so that it can be printed and read by someone else. I expect the same is true for video games - that a plan for the "plot" of the game is actually written, and all dialog is actually written just like it would be for a movie, etc. Where there is a team of writers, things get more complicated, which I will return to below.
After a project is written, then it must be turned into a form that can be experienced (unlike a novel, where the written form is the final form). Clearly, many people are called on to participate in the realization of the written ideas, but two jobs in a production have the blanket authority to override any and all writing done: directors and producers. Sure level designers might ask for a tweak in a story line or actors might ad lib some dialog, but it's usually a director who decides whether the change will or will not happen. In the music business (not counting film scoring or musical theater) the ultimate arbiter of what gets on a record is generally called a producer. For the other arts, the director has the final artistic say. On a TV show, a director does have authority of the episode(s) they are directing, but they can be overridden by the showrunner or producer if they want to do something that hurts the overall story line of the show. Technically producers can override directors, but the best ones know that doing so is a big risk to the quality of the final product. Someone has to be the final artistic line in the sand and generally that duty cannot be shared or compromised if the product is to be any good.
With that in mind, let's get back to writers. They write and then hand their results off to directors and producers and at that point, it's out of their hands, their work is mostly done, and they have to accept the loss of control over how the final product turns out.
Before that point, they are often working on a team. Many times, within a team of writers, there are also clearly defined duties. For example, musicals are composed of three written parts: the music, the lyrics, and the book or libretto (the equivalent of the script). Sometimes those three parts are written by three separate writers and they work together to make everything line up. Other times, there's collaboration on one or more component, or one person does music and lyrics, and the other does the libretto, to mention just one possible division of labor. For TV shows, there are often a team of writers led be a head writer and/or showrunner. This is a more structured situation where the showrunner alone determines the broad outlines of the story and then tasks the writing team with sorting out exactly what dialog will be used and actions taken by the characters to realize the plots and character development. This might be closest thing in creative writing to coding. Sometimes, individual writers will be in charge of a character, and advocate and inhabit that character. Sometimes there is a pitch session, where everyone on the team comes up with ideas that further the plot and pitches their ideas to the showrunner. The showrunner chooses the ideas that they think will work out the best and then everyone collaborates on that. The point is, there is still, in many ways, a separation of duties.
At the most microscopic level, sometimes two or more writers share the same role. This might be four band members all working on songs together (e.g., The Beatles), or a pair of people writing both the music and lyrics for a musical or collaborating on a screenplay.
In this situation, things are more complicated and fluid. Every writer as an individual generally has a "process" - the things they do to generate and refine ideas. When two or more writers really collaborate closely, they have to figure out how their processes can interact, and they have to find a new process which is the one that they all work through together. Just as every writer's process is individual and unique, so will every collaborative team's process be. When there are more than two people collaborating, even without a designated leader, often one or more leaders will emerge (e.g., John Lennon and Paul McCartney). That is probably simple group dynamics and not specific to writing. Whatever leaders exist or arise will be tacitly or explicitly serving as the equivalent of a showrunner. They will guide the overall flow of the work and often have veto power on ideas.
Are there guidelines such as "We only write sentences that are no longer than 15 words for character [x] because he wouldn't use that." and "Character [y] has to use the words 'Like, totally' at least two times per paragraph." for individual writers?
There are generally not guidelines that are that specific. But there will be guidelines that maybe are more artistic and less mechanical. Like, "During the course of this TV show, Dr. House will never be in a successful, happy romance for more than half a season". It's possible that a certain character might never use contractions. Often such decisions are made later in the process, by a director or even an actor or someone later in the process. Finster's accent/speech impediment in The Usual Suspects was entirely invented by Benicio del Toro (the actor who portrayed him).
On a TV show that hopefully will run for 100 or more episodes, it is much more important to have rules for the writing that will keep things consistent or changing in a deliberate way. Something like "the audience never gets to see what's inside the box". This is an area where someone from the writing team, i.e., the showrunner, gets to override a director. TV show directors come and go over the course of the show, but the story has to continue and continue to work.
How is a "Code Review" done? Is the lead writer doing all the editing so that the final product feels like it's written by one person and there are (not as many) incongruities?
Yes and no. A lead writer might actually write in any fixes, but my impression is they are more likely to send it back to the writing team with notes on what needs to be fixed and the team will fix it. At least in TV shows, there generally isn't a strong need for every episode to feel like it was written by the same person as every other episode. Watch enough TV and it becomes a lot more clear how the writing and directing varies from episode to episode. If a pitch for a whole episode is accepted from a staff writer, then often that staff writer is the "deputy showrunner" (my term) and the episode will have a tone from that staff writer. Other times a show runner or producer will "reward" a good staff writer with more leadership on an episode, or even a chance to direct an episode.
Two examples from my personal life:
I worked for a few years with another musician to write songs. Generally she came up with ideas and I was her editor, but I didn't just edit, I generated ideas also and she would critique my ideas and we would pick the best ones from both of us. She quickly became the leader regarding lyrics, and I ended up leading on the music. We talked many times about our goals for the music in general and for individual songs. Just as we each had to discover our individual processes, the process we used between us was something that we discovered and developed over the course of our collaboration.
Currently I'm working with one other person to write the book, lyrics, and music for a musical. We are both daunted by the task and a big part of our process is both being honest and realistic with each other on how much work it will be, while also maintaining each other's optimism about how good we think it can turn out, if we work hard. The overall story is my collaborator's idea, so I both had to decide that I was happy to tell that story with her, and then also defer to the story when I think I've got a good line or idea or whatever that doesn't help tell the story. In our process we are both coming up with ideas and discussing things at every level, from musical moods to whether a character knows what a certain word means. We have a lot of mutual respect and we both agree that the story is king. We may not always agree on what best serves the story, but we know that if and when it's clear something doesn't serve the story, it has to be cut, replaced, or refined. Basically we are each pitching ideas to each other and then acting as the other's showrunner - taking turns at both roles in a way.