Eu estudo melhor quando uso fones e não ouço o barulho da rua.

Yes. In Portuguese the subject pronoun is often omitted because the verb ending already shows who the subject is.

• Eu estudo melhor… and Estudo melhor… are both correct.
• Including eu can give a bit more emphasis on I (as opposed to someone else), or sound slightly more formal/explicit.
• In neutral, everyday speech, Estudo melhor quando uso fones… is very natural.

Melhor is the comparative form of bem (well) and also of bom/boa (good).

• bem = well
• melhor = better

Here you want to say I study better, not just I study well, so melhor is the natural choice.

Mais bem is possible in a few special cases, but with verbs like estudar, you almost always use melhor:

• Eu canto melhor. = I sing better.
• Eu durmo melhor. = I sleep better.

So Eu estudo melhor is the standard, idiomatic way.

Quando + present indicative (quando uso) is used for habits and general truths:

• Eu estudo melhor quando uso fones.
  = Whenever I use headphones, I study better (habitually).

Quando usar is the future subjunctive. That talks about a specific future situation:

• Quando usar fones, vou estudar melhor.
  = When I (eventually) use headphones, I will study better.

So, for a general habit, quando uso fones (present) is exactly right.

In this sentence, fones means headphones/earphones.

• It’s grammatically the plural of fone, but in real usage people almost always talk about these objects in the plural, just like English headphones.
• Singular um fone is understood, but sounds odd for headphones; you’d normally say it only if you really meant one earpiece or in some other specific context.

For European Portuguese, the more standard word is:

• auscultadores = headphones

So a very typical European Portuguese version would be:

• Eu estudo melhor quando uso auscultadores…

Fones is strongly associated with Brazilian Portuguese (from fones de ouvido). In Portugal:

• The “dictionary” word is auscultadores.
• Younger speakers may also say phones (anglicism), especially in informal speech.

European Portuguese speakers will usually understand fones, but if you want to sound more clearly European, prefer:

• Eu estudo melhor quando uso auscultadores…

Both are possible, but the meaning changes slightly:

• uso fones – generic: when I use headphones (in general).
• uso os fones – more specific: when I use the headphones (particular ones already known from context).

In your sentence, you’re talking about a general study habit, so the bare plural fones (no article) is very natural.

Ouço is the 1st person singular of ouvir (to hear):

• ouvir → eu ouço (I hear)

In European Portuguese, oiço is also accepted, though ouço is now more common in writing. Both are pronounced very similarly.

Escutar (eu escuto) exists too:

• ouvir = to hear (perceive sound)
• escutar = to listen (pay attention intentionally)

In practice, there’s a lot of overlap, but in your sentence:

• não ouço o barulho da rua = I don’t hear the noise from the street (the sound doesn’t reach me / is blocked).

That fits better than escuto here.

In Portuguese, the basic rule is:

• não + verb

So:

• Eu ouço o barulho. → Eu não ouço o barulho.

You don’t normally split não away from the verb in simple sentences. Variants like O barulho da rua não ouço are possible for emphasis, but they sound more marked and are less neutral.

Also, unlike some languages, standard Portuguese generally uses only one não for a simple negation here (no double negative with não + anything like não again).

O is the definite article: o barulho = the noise.

• não ouço barulho = I don’t hear any noise (noise in general).
• não ouço o barulho da rua = I don’t hear the noise from the street (a particular, known noise source).

Here you’re talking about that specific noise (street noise), so o barulho is natural.

Da is a contraction of de + a:

• de = of / from
• a = the (feminine singular)
• rua = street

So:

• da rua = de + a rua = of the street / from the street.

Portuguese almost always contracts these:

• de + a → da
• de + o → do
• em + a → na, etc.

So o barulho da rua literally is the noise of the street → the noise from the street.

Yes, but there’s a nuance:

• o barulho da rua = the noise that comes from the street (source).
• o barulho na rua = the noise in the street (location).

In context, both could describe similar real-life situations, but:

• não ouço o barulho da rua focuses on noise entering from outside.
• não ouço o barulho na rua focuses more on noise that is happening out there in the street.

Your original version is very idiomatic.

You can, and it slightly changes the focus:

• quando uso fones = whenever I use headphones (habit in general).
• quando estou a usar fones = when I’m in the middle of using headphones (focus on the ongoing action).

Both are grammatical in European Portuguese. For describing a general study habit, the simple present uso is more usual and more natural.

You can say it, but there’s a subtle difference in how it feels:

• quando uso fones e não ouço o barulho da rua
  First: I use headphones.
  Second: As a result (or at the same time), I don’t hear the street noise.

• quando não ouço o barulho da rua e uso fones
  First: I don’t hear the street noise.
  Second: And I use headphones.

The original order sounds more natural, because the headphones are the likely cause of not hearing the street. The re-ordered version sounds a bit less “cause–effect” and a bit more like just listing two conditions.