This is just reported speech and you can use either variant in both sentences. The difference is about how realistic you think the future action is. If you are unsure or certain it won't happen, you use 'would'. (This is always grammatically correct, by the way.) If you think the thing will definitely happen, you can use 'will'. It's similar to how will and would work in real and unreal conditionals. A simpler example is: 'John's just phoned to say he isn't coming to the office today. He said he 'is' ill.' - Using 'is' suggests that we believe the situation is true. 'John's just phoned to say he isn't coming to the office today. He said he 'was' ill.' - Using 'was' suggests that: we don't know that it's true; we don't believe that it's true; we think the situation may be different now. Tone of voice would also be very important here. So using will or would can show how confident we are in how true the thing you're reporting is or will be. The only thing is, if you know something is definitely untrue, you have to use the past form.

The first one is constructed that way because it's talking about "the future in the past." We're focusing of the fact that at the time he made the promise, he was talking about what was then the future. In your other example, we're focusing on the fact that the listening he promised to do is in the future compared to now. By the way, the word order would be more natural like this: He promised that he'll listen to English while he's driving tomorrow.

If you can use one of the words "will" or "would" in a sentence, then you can almost always use the other. The meaning, however, does change. Learning the subtle differences between modal verbs is one of the most important (and difficult) tasks that someone who wants to learn English needs to master. The primary role of the word "will" in English is to express conviction or certainty. Although it is often described as the key component of a "future tense", that importance is greatly overstated. In "he promised he will come tomorrow" "will" is used to express the firmness of a promise. It is not needed for "future" because the word "tomorrow" by itself does that very well: "he promised he is coming tomorrow". If you drop "tomorrow", then "will" can help to put the promise into the future: "He promised he will come". If you use "would" instead of "will", the effect is to lessen conviction and certainty, thereby introducing doubt or conditionality. "Would" also softens the sentence, making it less forceful. That is why "would" always seems more polite when you make a request: "Would you please pass the salt?" The sentence "he promised he would come tomorrow" is less precise than the one using "will". It could mean many different things. It could imply "he didn't come and broke his promise". It could mean "he stated he is coming tomorrow". It could mean "he might come tomorrow. At least, he promised". It could mean many things. Only context can provide an exact meaning. It is easier to say what it does NOT mean than what it does mean. It does not mean that he promised he will certainly come tomorrow.

The main difference between will and would is that would can be used in the past tense but will cannot. Also, would is commonly used to refer to a future event that MAY occur under specific conditions, while will is used more generally to refer to future events

Good question. You can also say "He did promise that he will come tomorrow" and you can also say "He promised he would listen to music while driving tomorrow". Although, the structures you used would tend to be worded how you used them by most educated people. This is not due to thinking about grammar logically. It's just due to habit. We just memorised phrases in this way. We don't really think about the logic involved, unless we're specifically linguistic scholars trying to explain it.