"I do not listen to a lot of music."
を changed to は because it's a negative sentence.
AFAIK (I'm not very proficient so it would be great if someone could verify this) は marks what you disagree with. There could be a question asking whether you listen to music daily. Then you could either mark daily with は and music with を, meaning that you don't listen to music daily, or you could mark music with は, meaning that you don't listen to music at all. Not having は seems to be just plain wrong here (i.e. you say no but at the same time agree with all the parts of the sentence).
You are making two questions there... first: why the sentence doesn't use たくさん instead of あまり, and second: why doesn't use the other kanji for 聴く
So let's start with the hardest one
"why あまり instead of たくさん？"
the original meaning of あまり is a little bit more complicated than just "a lot". Duolingo is just translating the Japanese sentence into English and teaching you the word. However you might need to be aware that あまり has a negative connotation, that's why is often used in negative sentences, and while you can use it in positive ones, the negative connotation still prevails.
Here is an extract from the DoJG:
The adverb あまり usually occurs in negative sentences, meaning 'not very (much)'. あまり is one of a group of adverbs which co-occur with negative predicates; In limited situations, あまり can be used in affirmative sentences, too. In this case, it means 'very; too' with a negative implication.
And the meaning of the kanji used for the word「余」has strong relevance to stretch, and it might imply to do something too much to the point of leaving only a remnant of resources. In Japanese, you use あまり as an adverb to soften the negative implication of the sentence. This is very similar to when you use "not much" in English, so I would personally translate this sentence into "I do not listen to music much", probably accepted by now in this exercise.
The reason why you don't use たくさん in here is because as I explained, the reason for あまり is not to mean "a lot" is just a softening expression from the language. Using たくさん here sound weirds to me here because of the negative meaning. たくさん would make more sense if the sentence is a positive one like たくさん書きます。Using it with a negative verb is like saying "I do a lot of not hearing" and that just sounds wrong.
Why 聞く instead of 聴く
聞 is a more general kanji, you can translate 聞く to "to hear", it doesn't have anything to do with music, while 聴 is more about paying attention while hearing and you can translate 聴く to "to listen" and in the context of music it also has an "enjoying" meaning.
In the exercise 聴く makes more sense to me, but Duolingo is probably using a more general kanji for this sentence because this is still a basic part of the course. Both are fine though.
Why is it は and not が? I swear... every time I think to FINALLY grasp the difference between the two I find out the rules changed, or there are additional rules or exceptions.
Here the 私は at the start of the sentence is understod but not included, right? Why is it then は and が is not accepted at all?
in this particular sentence is to place the focus of the sentence into the negation. The important part is あまり聞きません.
Since the topic is something that is known between both the listener and the speaker, at least conceptually, the topic is not important, what's important is what you are saying about the topic, then the rest of the sentence pops up in the listener's mind. The listener hears 音楽は "speaking of music" and he thinks "what about it?".. あまり聞きません。"I do not listen to it much".
If you use が、this would be ungrammatical since we expect を as in よく音楽を聞きます。