Estudio español en la biblioteca durante media hora.

In Spanish, subject pronouns are often omitted because the verb ending already shows who is doing the action.

• Estudio = I study
• The -o ending tells you it is first person singular: I

So Estudio español... is completely natural.
You could say Yo estudio español..., but yo is usually only added for emphasis, contrast, or clarity.

Estudio is the present tense, first person singular form of estudiar.

Conjugation of estudiar in the present:

• yo estudio = I study
• tú estudias = you study
• él/ella estudia = he/she studies
• nosotros estudiamos = we study
• vosotros estudiáis = you all study (common in Spain)
• ellos estudian = they study

Here, estudio means I study or I am studying, depending on context.

It can mean either.

Spanish present tense often covers both:

• I study Spanish in the library for half an hour
• I am studying Spanish in the library for half an hour

Which meaning is intended depends on context.
If you want to be very explicit about right now, Spanish can also use:

• Estoy estudiando español... = I am studying Spanish...

But the simple present estudio is very common and natural.

In Spanish, names of languages are not capitalized unless they begin a sentence.

So:

• Estudio español.
• Habla inglés.
• Aprendemos francés.

This is different from English, where you write Spanish, English, French with capital letters.

Also, español has the ñ, which is a separate letter in Spanish.

The written accent in español shows which syllable is stressed:

• es-pa-ñol

Without getting too technical, the accent mark helps indicate the correct pronunciation and stress pattern.

It is important to write it correctly:

• español = correct
• espanol = incorrect spelling

The ñ is also essential. It is not the same as n.

Because en means in or inside, while a usually means to.

• en la biblioteca = in the library
• a la biblioteca = to the library

So:

• Estudio español en la biblioteca = I study Spanish in the library
• Voy a la biblioteca = I go to the library

Here the sentence describes the location where the studying happens, so en is the correct preposition.

Biblioteca is a feminine noun in Spanish, so it takes the feminine singular article la.

• la biblioteca = the library

This is just the grammatical gender of the noun. It does not mean the library is biologically feminine; it is simply part of Spanish grammar.

A useful thing to notice:

• Many nouns ending in -a are feminine, though not all.
• Here, biblioteca is feminine, so la is correct.

Durante means for in the sense of throughout a period of time.

So:

• durante media hora = for half an hour

This phrase tells you the duration of the action.

You will often see:

• durante una hora = for an hour
• durante dos días = for two days
• durante la clase = during class

In this sentence, durante media hora explains how long the studying lasts.

Because hora is a feminine noun, and medio must agree with it.

• media hora = half an hour
• medio día = half a day / noon

So the form changes:

• masculine singular: medio
• feminine singular: media

Since hora is feminine, the correct phrase is media hora.

Yes. Spanish word order is flexible, especially with time expressions.

These are all possible:

• Estudio español en la biblioteca durante media hora.
• Durante media hora, estudio español en la biblioteca.
• Estudio español durante media hora en la biblioteca.

The most natural version depends on what you want to emphasize.

The original sentence is very neutral and natural:

1. action: Estudio
2. object: español
3. place: en la biblioteca
4. duration: durante media hora

Usually, no. When talking about school subjects or languages in a general sense, Spanish normally does not use the definite article.

So:

• Estudio español. = correct
• Aprende alemán.
• Enseñamos italiano.

Using el here would sound unusual unless you mean something more specific, such as a particular kind of Spanish or Spanish as an academic field in a special context.

For a general statement, español without an article is the normal choice.

Yes, it is natural and correct in Spain.

A speaker from Spain would understand it as a normal sentence meaning that the speaker studies Spanish in the library for thirty minutes.

A few natural alternatives might also be heard, depending on style:

• Estudio español en la biblioteca media hora.
• Estoy estudiando español en la biblioteca durante media hora.
• Estudio español en la biblioteca durante treinta minutos.

But your original sentence is perfectly good Spanish.