^ Once a teacher at my school like 5 years ago DID write a letter of rec that essentially said "I wouldn't accept this kid if I were you. He causes trouble, doesn't hand in homework, and steals." But apparently that kid went to the teacher like two days before the letter had to be sent in, and the teacher already heavily hinted that he didn't want to write the rec for him but the kid begged him anyways. So yeah, teachers can write negative recs for students, and that would automatically equate to rejection most of the times once an adcom reads it. But teachers usually don't do that so I guess you shouldn't worry OP.
1) How big of an impact does teacher recommendations have on your admission chances?If it's a very neutral type of letter that just generally lists off things you're general things you're good at without any personal details from the teacher, then the letter doesn't really help you (but it won't be a bad impact either). A good letter would talk about specific examples of your strengths in the class or struggles you've overcome in the class. Teachers who write good recs would also ask for your resume so they can describe you even more personally, like they know what you do outside of class (and also have a chance to reaffirm any good qualities about you outside and inside the classroom; i.e. hardworker, or leadership qualities). If you were to be compared with another college applicant who was like you in every way, except that you had way better teacher recs, then you would be accepted first because you have teachers vouching for who you are. Of course, a negative rec from a teacher can almost equal an automatic rejection.
So I would really recommend that when you ask your teacher for a recommendation, first ASK. It really isn't hard at all, you just go up to them and politely/cheerfully ask "Mr./Ms. ____, I just wanted to ask if you would be comfortable with writing a recommendation letter for me to college?" This gives a teacher the leeway to say no if they don't feel good about writing one for you. Don't feel bad if they say no, it just means that they don't know you well or are too busy. If they say yes, ask them if they need any materials to help them better write your rec letter. Then promptly get the materials to the teacher in a folder within a couple days. The folder should have your name, and the inside the typical papers in it are: list of colleges you're applying to and their deadlines for teacher recs, stamped envelopes with addresses of each college written, the common app teacher papers for each college, and your resume.
2) Is it 'inappropriate' to ask a teacher for a recommendation when I didn't particularly "participate enthusiastically" in their class? In all honesty, I don't participate in any class, really. What do all the other quiet kids do?No way! In my gov class, more than 3/4 of the class never says anything - because his class is mainly lecture-style with mainly side comments from students who bother to keep up with politics. And the 3/4 of us that usually don't say anything aren't as politically informed, so we don't bother. (I'm in that 3/4 of the class.

) But the gov teacher still had 20 rec letters lining up for him to do, and he accepted to do them all. And in classes like math, there is no way to participate beside answering the occasional math question.
As long as you do all the teacher's work, don't act out in their class, and get fairly good grades in their class, then the teacher shouldn't have issues writing a letter for you usually.
I'm usually a pretty quiet kid in most classes too! But my English teacher still loved me because I wrote well for all essay assignments, and because I always did nice artwork for the big English projects haha. As for my calculus AB class, I actually did horrible in the class last year. Got failing grades on a lot of the quizzes/tests, and only got good grades on the take-home assignments and wrote problems on the board almost everyday. I got B's the first 2 quarters and A's the last two quarters on my report card. (Actually, I had to kinda plead my math teacher to make my 4th quarter grade an A since it was a 88 lol!) And this same teacher whose class I did miserable in wrote a rec letter for me that won me a full paid scholarship to any college. :] She's actually a very nice teacher though.
3) Which teachers should I approach?It depends on what major you're applying for in college. Typically, you ask one rec from a science/math teacher and one rec from a social science/liberal arts subject. But of course, ask a teacher who you think best knows you. Or ask the teacher who is super nice and bound to write something good about you if you haha. I have a Spanish teacher who is like your kind grandmother that only has nice things to say about everyone. But I prioritized my english teacher who knew me personally better, and my math teacher who can talk about me overcoming my struggles in math.
In college application cases, it seems suspicious and dodgy to adcoms if you for instance do bad in math, and don't have any rec letter from a science or math teacher. Because it might show that you have a weakness in science/math subjects. I would say, don't be afraid to ask a teacher to write a rec even if you don't do the best in their class. If that teacher sees that you still try hard, then they will probably still say something good about you. Of course, it is still best to get a rec from a class where the teacher knows you well AND you do well in their class.
I'd suggest you not go with a letter from your art teacher.... She seems to not know you very well. Unless you do really great art work in her class, and you're applying to an art school or as an art major, then getting a rec from an art teacher doesn't really benefit you much. Unless they know you very well, convey that in the letter, and speak highly of you. Just because your language teacher seems to be deducting a lot of points off from you, maybe she is just trying to make sure you don't make the same mistake again? However, I don't know your situation on that in depth so I can't really say.

Since you're still early in junior year, it's easily possible for you to just start participating more in class. (It's not even January yet.) Look for chances to raise your hand. Like wait for the easy or no-duh questions, and shoot your hand up to answer them. Just raising your hand once or twice every week will make your presence definitely known to the teacher. If the teacher assigned anything with a long-donkey explanation in class, wait till after class to ask them any questions you have about the assignment to confirm you heard/understood it correctly. It'll show that your dedication to want to do the assignment well. Or whenever you don't understand something being taught in class, ask the teacher in the early morning, during lunch, or after school. Come with specific questions to ask, not general ones. (i.e. ask how to do numbers 20 and 53 on page244, not ask something like "I don't understand the calc stuff in Chapter 7, how do you do it all?") When you ask teachers for help, it shows that you are at least trying to make an effort to improve in their class. Only meanie teachers would turn away a student who needs help.
I just sent in my ED app to a college 2 days ago, and I got my rec letters from my English and Calc AB teacher from 11th grade. So do the folder thing! Half the teachers in my school basically require it, along with the resume. Hope my long droning post helped you in some way. XD