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I study English while I'm listening to music.
I study English listening to music.
These two looks like putting emphasis on studying English a little more than listening to music
I study and listen to music at the same time.
It's like this sentence equates and coordinates two clauses equally.
These 3 sentences are perfectly replaceable?
Yes and no. I think Brian Hitchcock's comment just gave me the crucial clue.
These examples should clarify what's going on:
(1) I do everything while listening to music! I wash the dishes while listening to music, I jog across campus while listening to music, and I repair cars while listening to music. I even study English listening to music and sleep listening to music.
(2) Music is the greatest pedagogical tool known to humanity. You can learn anything by setting it to music, listening to it, and singing along with it. After a few days, you'll never forget it! I learned the quadratic formula in algebra from a song with the same melody as "Twinkle Twinkle Little Star". Later, I learned chemical thermodynamics just by listening to music. Currently, I study English listening to music and seismology watching interpretive dance.
That the first example means "while listening to music" and the second example means "by listening to music" are both completely clear and unambiguous even though the second example is absurd.
Here's what's happening. "Listening to music" functions as an adverb modifying "study", but the fact that "listening" is a participle makes a listener's mind search the sentence for a reasonable noun to attach it to. It gets attached to "I". So now the mind understands the sentence as describing two actions performed by "I": studying and listening. The main verb is "study", and "listening" somehow connects with it. The sentence must be asserting some relationship between the two actions. What is that relationship? Study while listening? Study by listening?
If the preceding sentences already establish such a relationship, the mind is "primed" to find that relationship very easily and not to find other reasonable candidates.
Here is the important grammatical principle: because "listening to music" lacks a subject and lacks a preposition to explicitly connect it with the verb, it is extremely flexible with regard to those missing items. Consequently, it is also very ambiguous on its own. Different native speakers understood "I study English listening to music" in your original question in very different ways, or found the sentence ambiguous or unclear. So, a sentence like that needs context to indicate the relationship between "listening" and "study", but when that context is provided, the relationship can be clear.
By the way, this construction is somewhat unusual in English but very common in the Romance languages. In the Romance languages, it has the same ability to represent actions that occur together or one describing how the other is accomplished. For example, in Spanish:
Estudio Inglés escuchando música. [Source]
Estudiando mucho, tendremos éxito. [By studying hard, we will be successful. Source]
In Germanic languages, usually a verb for motion indicates the manner of motion, and the direction of the motion is indicated by a preposition or particle: "run in", "walk out", "crawl through", "bike to". In Romance languages, usually a verb for motion indicates the direction of motion, and the manner of motion is indicated by an adverbial phrase like the construction you're asking about: salir correndo (exit running), llegar en bicicleta (arrive on bicycle), entrar sonriendo (Enter Laughing). Linguists call these "satellite framing" and "verb framing", respectively.
Notice, though, that English does include verb-framing constructions: the ones I put in parentheses, which literally translate the Spanish into perfectly good, though unusual, English. I don't know for sure, but this might be a case of English incorporating foreign grammar just as it incorporates foreign vocabulary. Supposedly this construction is more common with verbs from Latin. Maybe it's a coincidence that you asked about "I study English listening to music" rather than than "I learn English listening to music"; maybe not. ("Study" is from Latin; "learn" is Germanic.)
Regardless, your question and the responses to it probably well illustrate the principle that English is a "mutt" language, where multiple, different, somewhat incompatible vocabularies and grammatical principles coexist and sometimes compete. You shouldn't look for consistency across all of them, but within each there is some consistency.
No not really.
I study English while I'm listening to music.
This one is pretty clear. You are listening to music at the same time as you are studying English.
I study English (by) listening to music.
This means that listening to music is helping you learn English (the lyrics, I suppose). There is no while or same time here.
I study and listen to music at the same time.
This has the same meaning as the first sentence.
