Learn Hardcore Malay: Rak buku di asrama saya sudah penuh dengan buku sejarah dan sains. - The bookshelf in my dormitory is already full of history and science books.

Here penuh (full) acts like a verb/adjective meaning “is full”. You only use adalah/ialah mainly in more formal contexts to link two noun phrases, e.g.:

Ali adalah seorang doktor. – Ali is a doctor.

You would not normally say: Rak buku di asrama saya adalah penuh – that sounds unnatural in standard Malay.

What is the function of sudah before penuh? Can I leave it out?

Sudah is an aspect marker meaning “already” or indicating that a state has been reached or an action completed.

Rak buku … sudah penuh – The bookshelf is already full / has become full.

If you leave out sudah:

Rak buku di asrama saya penuh dengan buku sejarah dan sains.

This simply states a fact: The bookshelf is full of history and science books.
Using sudah adds the nuance that it wasn’t full before, but now it is.

What’s the difference between sudah, telah, and dah?

All three can express the idea of “already” or completed action/state, but they differ in register and frequency:

sudah – neutral, very common in both spoken and written Malay.
telah – more formal/literary, often used in news, official writing, or formal reports.
dah – colloquial/shortened form of sudah, very common in informal speech.

In your sentence, you could say:

Rak buku … sudah penuh … – neutral, standard.
Rak buku … telah penuh … – more formal.
Rak buku … dah penuh … – sounds casual/colloquial.

Why do we say “penuh dengan” instead of just “penuh”?

Penuh means “full”, and dengan here introduces what it is full of:

penuh dengan buku sejarah dan sains – full of history and science books.

In careful standard Malay, with this meaning, penuh is usually followed by dengan + noun phrase.

You might hear people say in casual speech:

Rak buku saya penuh buku sejarah dan sains.

This is understandable, but penuh dengan is more standard and natural in writing and in careful speech.

Does dengan here mean “with” in the same way as “together with someone”?

It’s the same word dengan, but the function is slightly different.

Saya pergi ke pasar dengan kawan.
I went to the market with a friend. → “together with”
Rak buku … penuh dengan buku sejarah dan sains.
The bookshelf is full of history and science books. → “full of” / “filled with”

So in this sentence, dengan is part of the fixed pattern penuh dengan X = “full of X”, not “together with X”.

What is the difference between “di asrama saya” and “dalam asrama saya”?

Both di and dalam can be translated as “in/at”, but:

di = at/in (more general location marker)
dalam = inside (more strongly emphasizes interior/inside)

In this sentence:

Rak buku di asrama saya … – The bookshelf in my dorm … (natural, neutral)
Rak buku dalam asrama saya … – feels like you’re stressing inside the dorm building, which is usually unnecessary unless you’re contrasting “inside vs outside”.

So di asrama saya is the most natural choice here.

Could the word order be “Rak buku saya di asrama sudah penuh…”? Is that correct?

Yes, that is grammatically correct, but the focus shifts a bit.

Rak buku di asrama saya sudah penuh…
– Slight focus on where the bookshelf is (the one in my dorm).
Rak buku saya di asrama sudah penuh…
– Slightly stronger sense that it’s my bookshelf (and it happens to be in the dorm).

In everyday usage, both are acceptable and mean almost the same. The original version (di asrama saya right after rak buku) sounds very natural and common.

What exactly does “rak buku” mean? Is it a bookshelf, a bookcase, or something else?

Rak = rack/shelf
Buku = book

So rak buku literally means “book rack / book shelf.” It’s a general term and can cover what English calls:

a bookshelf (open shelves),
a simple book rack,
sometimes even a small bookcase, depending on shape and context.

For a more cupboard-like furniture piece with doors, you might also hear:

almari buku – book cupboard/bookcase.

But in many everyday contexts, rak buku is what people say for “bookshelf.”

Why don’t we mark “buku sejarah dan sains” as plural, like “books”? There’s no “-s” or anything.

Malay generally does not mark plural with endings like -s. The noun buku can mean:

buku = book / books (depending on context)

Plurality is understood from:

context, or
quantifiers (e.g. banyak buku – many books),
or reduplication (buku-buku) in some cases.

Here:

penuh dengan buku sejarah dan sains
logically implies many books, so buku is understood as “books” in English.

What exactly does “buku sejarah dan sains” mean? Is it “history and science books” or “books about both history and science”?

By default, buku sejarah dan sains is understood as:

“history books and science books.”

That is, buku sejarah (history books) and buku sains (science books) are being joined.

If you really wanted to say “books about both history and science together” you’d usually rephrase more clearly, for example:

buku tentang sejarah dan sains – books about history and science
buku mengenai bidang sejarah dan sains – books about the fields of history and science

But in normal everyday interpretation, buku sejarah dan sains = history and science books.

Could we drop the first “buku” and just say “penuh dengan sejarah dan sains”?

You can say penuh dengan sejarah dan sains, but that changes the meaning:

penuh dengan buku sejarah dan sains
→ full of history and science books.
penuh dengan sejarah dan sains
→ full of *history and science (as abstract subjects/concepts).*

The second one sounds more abstract or metaphorical, like saying “my life is full of history and science.”
To make it clear you are talking about physical books, it’s better to keep buku:

penuh dengan buku sejarah dan sains.

Is “asrama saya” the only way to say “my dorm,” or can I say “saya asrama” like in English word order?

In Malay, possessive pronouns like saya come after the noun:

asrama saya – my dorm
buku saya – my book
kereta saya – my car

You cannot say saya asrama to mean my dorm; that would sound wrong.

You might sometimes hear a borrowed word like dorm used informally:

dorm saya – my dorm (mixing English and Malay)

But within standard Malay, the correct order is noun + saya, so asrama saya is the proper form.