Studio in biblioteca da mezz’ora.

Because Italian commonly uses the present tense + da + length of time to express an action that started in the past and is still continuing now.

So:

Studio in biblioteca da mezz’ora.

means something like I have been studying in the library for half an hour.

This is different from English, which usually needs have been studying. Italian does not need a special tense here; the simple present does the job.

A very common pattern is:

present tense + da + time expression

For example:

• Abito qui da due anni. = I have lived / I have been living here for two years.
• Aspetto da dieci minuti. = I have been waiting for ten minutes.

Here da means for in the sense of for the last amount of time up to now.

So da mezz’ora means:

• for half an hour
• more literally, since half an hour ago

In this structure, da introduces the duration of an action that is still in progress.

Compare:

• Studio da mezz’ora. = I have been studying for half an hour.
• Studio dalle tre. = I have been studying since three o’clock.

So da can mean either:

• for
  • a duration
• since
  • a starting point

Mezz’ora is the usual contracted form of mezza ora.

This happens because mezza loses its final vowel before ora, and an apostrophe marks that omission:

• mezza ora → mezz’ora

Both relate to half an hour, but mezz’ora is the normal form you will most often see.

You also see similar forms such as:

• un’ora from una ora
• l’amico from lo amico (historically; modern standard is l’amico)

Because Italian usually does not need subject pronouns when the verb ending already makes the subject clear.

Studio by itself means I study / I am studying.

The ending -o tells you the subject is io.

So:

• Studio = I study / I’m studying
• Studi = you study
• Studia = he/she studies

You can add io for emphasis or contrast:

• Io studio, tu lavori. = I’m studying, you’re working.

But in a normal sentence, leaving it out is more natural.

It can mean both, depending on context.

The Italian present tense often covers:

• habitual actions: I study
• actions happening now: I am studying

In this sentence, the phrase da mezz’ora makes it clear that the action is ongoing right now, so studio is understood as:

• I’ve been studying
• or in a more literal present sense, I am studying

So the time expression tells you how to interpret the verb.

In biblioteca is the normal way to say at/in the library when you mean the place in a general sense.

Italian often omits the article in expressions with certain public places or institutions:

• in banca = at the bank
• in chiesa = in church
• in biblioteca = in the library

If you say nella biblioteca, it usually sounds more specific, as if you mean in that particular library / inside the library building.

So:

• Studio in biblioteca. = I study at the library.
• Studio nella biblioteca dell’università. = I study in the university library.

Yes, you could, and it is grammatically correct.

Sto studiando is the progressive form, similar to I am studying. So:

• Studio in biblioteca da mezz’ora
• Sto studiando in biblioteca da mezz’ora

Both can work.

However, in Italian, the simple present is often enough, and it is very common in this kind of sentence. The progressive form is possible, but not always necessary the way it often is in English.

So the version with studio sounds very natural.

Yes, Italian word order is fairly flexible.

These are all possible:

• Studio in biblioteca da mezz’ora.
• Studio da mezz’ora in biblioteca.
• Da mezz’ora studio in biblioteca.

The meaning stays basically the same, but the emphasis changes slightly.

For example:

• in biblioteca earlier in the sentence highlights the place
• da mezz’ora earlier highlights the duration

The original sentence sounds natural and neutral.

In this sentence, no. Here studio is clearly the verb studiare: I study / I am studying.

That is because it is followed by:

• a place: in biblioteca
• a duration: da mezz’ora

Those elements describe an action in progress.

If you wanted study as a noun in English, Italian would use a different word, such as:

• lo studio = the study, the office, the act of studying depending on context

So here studio is definitely the verb.

The pattern is:

present tense verb + place + da + duration

In this case:

• Studio = present tense verb
• in biblioteca = place
• da mezz’ora = duration continuing up to now

A very useful formula to remember is:

Italian present tense + da + time = English have been doing / have done for

Examples:

• Lavoro qui da un anno. = I have been working here for a year.
• Ti aspetto da cinque minuti. = I have been waiting for you for five minutes.
• Viviamo in Italia da poco. = We have been living in Italy for a short time.

No. If the action is finished, Italian would not normally use this present-tense pattern.

This sentence works because the action is still continuing now.

If the action were over, Italian would use a past tense, depending on the context. For example:

• Ho studiato in biblioteca per mezz’ora. = I studied in the library for half an hour.

Notice the change:

• da mezz’ora with the present = action still going on
• per mezz’ora with a past tense = action lasted half an hour and ended

This is an important difference:

• Studio da mezz’ora. = I have been studying for half an hour.
• Ho studiato per mezz’ora. = I studied for half an hour.