There may be some confusion about what the different roles on a game development team really are.

Writing plot points and what not is not so much design work as it is writing work. In some companies that falls under the Design umbrella but others have entirely separate Fiction departments. A game designer generally has more impact on the actual gameplay while a writer focuses on the narrative and explaining the "why" when it comes to using or justifying the game mechanics.

Artistic skill has very, very little to do with design; designers are not expected to produce any art. Games have dedicated art teams to deal with the actual art. Designers focus on the actual game, be it core mechanics, UI flow, monetization strategies, level design, and all the grunt work in actually balancing and polishing the game. And level designers might be the closest to being artists than the other sorts of designers, not the furthest as you implied.

Engineers still have creative input. Very large companies might drown out engineers as nothing but code monkeys, but in any competent (even pretty large) company it's well understood that making games is a cooperative effort that requires input from everyone, from QA (especially QA, really) through engineering to art to design to production and so on. At some companies you might even find that an engineer has a bigger creative impact than most designers by simple virtue of there being more designers on staff than there are engineers. Coding itself can be quite creative, too. And in some organizations you may find that engineers are expected to help drive some of the core gameplay features that they're responsible for implementing, especially if there's few technical or systems designers on staff.

In very small companies, you'll wear many hats. All of the engineers might be expected to do double duty as designers and QA or the designers may well also be the artists and producers.

In both small and large companies you'll also find roles that straddle bridges. Gameplay engineers touch a lot of design; technical designers and technical artists can touch a lot of code. Test engineers and technical QA can write code and help drive gameplay based on quantifiable feedback.

If you're in a company where a good idea is shot down just because it came from an engineer, or a designer is handicapped from being able to implement the solutions they need to try out new ideas, or a test is just treated as a monkey who fiddles with the game and enters bugs into Jira, you've found yourself in a terrible game company. Unfortunately, there are many of those.

In general, do what you like. If you go the coding route and hate it, you're going to be a bad hire that nobody wants to keep on the books. If you go design simply because you think it's more creative and not because you actually like the work, you'll likewise be a bad hire. Do what you enjoy doing. If you enjoy doing a few different things, try your hand at all of them; make a small (small!) indie game to try them out.

If you like writing, stick to writing. Get a traditional journalism degree. You might end up being a fiction writer for a game - or even a researcher for a more historically based game - or end up writing traditional books, or writing TV scripts, or so on; I wouldn't fixate too much on games until you know what it's really all about; making games is probably nothing at all like you think.

1) From what I understand, the designers do most of the creative work, putting characters, maps, and levels together, and the developers mostly translate it into coding and put it all together.
2) My only problem with being a designer is that I have no artistic ability beyond stick figures. Could I succeed in a game designing career without artistic skills?
3) On the other hand, I don't know if I want to be a full-time coder either ... coding for 8 hours a day, 5 days a week would get too tedious.
4) I definitely want to have some creative impact on my games' development -- Is there a way that I could take on both roles, help with the coding and design?
5) Should I just stick to writing and consult with the designers so I can get my ideas put into the games?

1) Nope, you're mixing together a LOT of different jobs:
Game developer -- anyone who works for a games company and works on the game. This includes designers!
Game designer -- is an expert at talking about game mechanics. Can design a board game that's actually fun. Also often has to do a lot of similar work to a Producer / Project Manager, in order to make sure everyone else is effectively moving forward on the project. Also, these people make up about 1% of the whole team they're rare -- e.g. you might have 50 other staff for each game designer.
Level designer -- knows a lot about game design, and how spaces affect gameplay. They work hand-in-hand with the 3d-environment-artists and game-designers to design the spaces in which the game will take place. After they've designed a space, the 3d environment artists will make it look pretty.
Game programmer -- writes the code that makes all the things in the game happen. Sometimes they do exactly what the game designer says to do, other times they have a lot of freedom to interpret and iterate on the the designer's original ideas.
Concept artist -- draws illustrations to guide the whole team, helping them visualise the end-product before it's been created. Often are the ones who design the 'look' of the characters/environments during the pre-production phase.
Environment artists -- make the pretty art/models/textures for buildings, locations, worlds.
Character artists -- sculpt the characters for the game.
Texture artists -- sometimes studios have dedicated people who's whole job is to paint the surfaces of 3D objects created by other artists.
Rigger -- takes the characters (and other moving objects) and attaches them to a skeleton so they can be animated.
Animator -- takes the rigged characters/other objects and creates all the different animations required, such as walking/running/jumping/etc... Sometimes using mo-cap data as a base.
Writer -- not usually a full-time job at a studio. Writes the storylines, etc... but this has zero impact on the game mechanics.

2) Designers produce zero artwork, so you're fine there.
3) If you don't like the idea of doing one job for 8 hours a day, then the workplace is not going to be fun
4) At some studios, game-programmers have a lot of input when translating the game-designers' mechanic ideas into reality... but at other studios you don't have any creative input. If you don't enjoy the creativity of writing code itself, you might not want to be a coder...
Going the "indie" route lets you be responsible for every single role in a company though
5) Being a writer is a completely different job to being a game designer.