Eu estudo português durante quinze minutos.

Yes. In Portuguese it’s very common to drop the subject pronoun when the verb ending already shows who the subject is.

• Eu estudo português durante quinze minutos.
• Estudo português durante quinze minutos. ✅ (also correct)

Both mean the same thing: I study Portuguese for fifteen minutes.
Using Eu adds a bit of emphasis on I, but grammatically it’s not required.

Estudar is the infinitive form: to study.

Estudo is the 1st person singular, present tense:

• eu estudo = I study

Very simplified pattern for estudar (regular -ar verb, present tense):

• eu estudo – I study
• tu estudas – you (singular, informal, PT) study
• ele / ela / você estuda – he / she / you study
• nós estudamos – we study
• vocês / eles / elas estudam – you (pl.) / they study

In the sentence, estudo matches eu.

On its own, Eu estudo português durante quinze minutos. is naturally understood as a habit or routine:

• I (usually) study Portuguese for fifteen minutes.

For something happening right now in European Portuguese, you’d normally say:

• Estou a estudar português há quinze minutos.
  = I have been studying Portuguese for fifteen minutes (up to now).

or

• Estou a estudar português durante quinze minutos.
  = I am going to study Portuguese for fifteen minutes (right now).

In Portuguese:

• Languages are written with a lowercase initial:

  • português, inglês, francês

• Nationalities used as adjectives or nouns are also lowercase:

  • sou português = I’m Portuguese
  • um aluno português = a Portuguese student

You only capitalize Português in things like Português (school subject title) when all subjects are capitalized by convention (similar to headings).

After verbs such as estudar, aprender, falar etc., the language name is usually used without an article:

• Eu estudo português. ✅
• Eu falo português. ✅

Eu estudo o português. is grammatically possible but sounds unusual in everyday speech. O português is used more when you mean the Portuguese language as an object of discussion:

• Gosto do português. = I like Portuguese (the language).
• O português é uma língua difícil. = Portuguese is a difficult language.

Yes. In this context durante corresponds closely to English for (duration):

• durante quinze minutos = for fifteen minutes

It simply states how long the action lasts.

You can often also use por in a similar way:

• Eu estudo português por quinze minutos. ✅ (also acceptable)

In everyday speech, people sometimes even omit the preposition:

• Eu estudei português quinze minutos. (colloquial, more common in speech than in writing)

But durante is a clear, standard choice.

Yes, adverbial time expressions are quite flexible. All of these are possible:

• Eu estudo português durante quinze minutos. (very neutral)
• Durante quinze minutos, eu estudo português. (more emphasis on the duration)
• Eu, durante quinze minutos, estudo português. (more marked; adds focus, less common in simple sentences)

What you normally don’t do is place the duration between the verb and the direct object in this case:

• ✗ Eu estudo durante quinze minutos português. (sounds awkward)

Because quinze (fifteen) is greater than one, so the noun goes in the plural:

• 1 minuto – one minute
• 2 minutos – two minutes
• 15 minutos – fifteen minutes

General rule: numbers greater than one take the plural form of the noun.

Yes, you can:

• Eu estudo português durante um quarto de hora.

This is understandable and correct, but in modern everyday speech people more often say quinze minutos rather than um quarto de hora, especially in Portugal. Um quarto de hora can sound a bit more formal or old-fashioned, depending on context.

• estudar = to study, to put in effort, to work on something
• aprender = to learn, to acquire knowledge/skill

So:

• Eu estudo português durante quinze minutos.
  = I study Portuguese for fifteen minutes.

If you say:

• Eu aprendo português durante quinze minutos.

it sounds odd, because learning is more the result than the activity. You normally say:

• Eu estou a aprender português. = I’m learning Portuguese.
• Eu estudo português para aprender mais. = I study Portuguese to learn more.

Approximate European Portuguese pronunciation (very roughly):

• Eu → like “ehw” / “ayo” (closed sound, one syllable)
• estudo → “sh-TOO-doo” (initial e often reduced: /ʃˈtu.du/)
• português → “por-tu-GESH” (final ês like “esh”)
• durante → “doo-RAN-t(ə)” (final e often very weak /ə/ or almost silent)
• quinze → “KEEN-zə” (the qu is like k, final e weak /ə/)
• minutos → “mee-NOO-tush” (final os often pronounced /uʃ/)

Spoken smoothly, it’s something like:

• [ew ʃˈtu.du puɾ.tuˈgeʃ duˈɾɐ̃.tɨ ˈkĩ.zɨ miˈnu.tuʃ]

Using estudar in other simple tenses:

• Past (completed):

  • Eu estudei português durante quinze minutos.
    = I studied Portuguese for fifteen minutes.

• Near future (going to):

  • Eu vou estudar português durante quinze minutos.
    = I’m going to study Portuguese for fifteen minutes.

• Habitual past (used to):

  • Eu estudava português durante quinze minutos todos os dias.
    = I used to study Portuguese for fifteen minutes every day.

They express different ideas:

1. durante quinze minutos = for fifteen minutes (duration of the action)

   • Eu estudo português durante quinze minutos.
     = I study Portuguese for fifteen minutes.

2. há quinze minutos (in European Portuguese) = fifteen minutes ago
   or, with há … que, for fifteen minutes (up to now):

   • Cheguei há quinze minutos.
     = I arrived fifteen minutes ago.

   • Há quinze minutos que estou a estudar português.
     = I’ve been studying Portuguese for fifteen minutes.

So in your original sentence you need durante because you’re talking about how long the action lasts, not when it started.