1: If I was going to be the Art Director (and maybe also create assets), coming in at the very beginning would be fine, to start brainstorming and such. For someone making assets that are supposed to ship with the game, I'd hope that the art style had already been clearly determined, or I would fear that my work might all get thrown away eventually, or that the art assets won't all match each other.

2: At the beginning would be great, or by 'the middle of pre-production' at the latest.

I think a lot of small Indie projects skip over 'defining the art style', which is a problem for many games. As long as defining the art style is something that is going to be done, starting that process early on is great.

Also, for some games, the backstory and setting need to be nailed down to some degree before an art style (and specific art assets) can be created. I once worked on a 'AAA' title that was supposed to be set in the year 2000 roughly (this was a couple of years in the future, at the time), and then, about the time the project was supposed to be in Beta, the backstory was changed to be around the year 2080. The art had already been made. After a while, the studio was shut down and the project canceled...

1) As an artist, how much of a game's style and/or setting would you want to be established (and, I suppose, how established) before you started on the artistic design (visual, mostly, but I suppose this could also extend to music and sound-design)?

None, I find that it's harder to create good graphics when you are working on a unstable foundation.

If some kind of style or setting is important to the design, the developer should tell me and provide feedback on the updates I provide.

2) How early in development of the game (it's small enough that it's not really using an engine, so what would normally be considered engine-development would also fall under game-development), in terms of mechanics and scope, should the artistic design be started? I could imagine that while mechanics might influence the art, art might also inspire changes or addition/elimination of mechanics (maybe this isn't the case, though)

While true the theme and style doesn't need to be decided before you start the game, the first God of war is a good example and many AAA games change style as they develop, it helps a great amount if these things are decided before you start the game.

The main problem is that unless your 3D modelers or 2D artist know what they are doing, the game kind of grinds to a hold. The 3D modelers can create some place holder stuff however no animator is going to spend the days working on animating a placeholder, they will also just do some thing very basic or show there own ideas; same for 2D.

With out a proper theme when starting you will be paying your artist full pay for light work.

The approach most indie developers take is also good, making the game with programmer art and then hiring a artist to do the final art. The thing to note is that the game will change, no matter how much you attempt to stick to the original design.

Your artist and level designer will move things around to match the style of the game or to create atmosphere, the best programmers I worked with incorporate these changes into the game making it work for them.

1) As an artist, how much of a game's style and/or setting would you want to be established (and, I suppose, how established) before you started on the artistic design (visual, mostly, but I suppose this could also extend to music and sound-design)?

It's impossible for me to have good ideas for designs if there isn't some framework to start from. This means a context (universe of the game) and plot (what happens during the course of the game).
Supplemental material like moodboards and soundtracks (from other games and film for example, anything that conveys the moods you want) can help explain your vision for the game.

2) How early in development of the game, in terms of mechanics and scope, should the artistic design be started? I could imagine that while mechanics might influence the art, art might also inspire changes or addition/elimination of mechanics (maybe this isn't the case, though).

To me the game design and art are tied, so one works off the other. If you want something that feels meaningful, plan the game design and art together. The art may evolve with time, but whenever you think of "player character" there should be something in your mind of what they look like -- the latest revision of their design.

Since it's a 2D sprite-based game, I know at least the limitations inherent to that art-style, but I haven't really done anything beyond implementing simple sprites and animations, and I'm not sure what features would extend what I have in a meaningful way that an artist would really be able to appreciate and take advantage of.

It depends on the artist. For me, being able to tell an entertaining story and have appealing characters is paramount, so if I was supposed to be excited about working in your project I would want something like that. Even if it's a puzzle or match-3 game, having some kind of cinematic approach.