Learn Hardcore Japanese: tosyokan ni ha hon ga ooi desu. - There are many books in the library.

If you said only 図書館に本が多いです, it could still be understood, but it sounds less natural and loses that “as for the library” topic feeling.
In everyday speech, 図書館には is the most natural way to start this sentence.

Is には one single particle, or always に + は?

Grammatically, には is always に + は:

に = location, time, direction, etc.
は = topic (often with a contrastive nuance).

They often appear together, so you see には written as one block, but they are still two particles, each doing its own job.

The same is true for combinations like:

では = で + は
へは = へ + は

In all these, the first particle marks some basic role (place, direction, etc.), and は then topicalizes that phrase.

What exactly does が do after 本? Why not は or another particle?

In 本が多いです:

本 is the thing being described.
が marks it as the grammatical subject of the description.

So 本が多いです literally means:

Books are many (i.e. There are many books).

Why が, not は?

が marks what is many.
は would mark the topic “as for books…”.

Compare:

図書館には本が多いです。
As for the library, (the) books are many. → There are many books in the library.

If you said 図書館には本は多いです, that would suggest a contrast like:

In the library, *books, at least, are many (maybe other things aren’t).*

So here, が is the natural choice: it simply marks 本 as the subject of 多い.

Where is the verb in this sentence? Japanese is SOV, so shouldn’t there be a main verb like あります?

In Japanese, adjectives can function as predicates, so an adjective can work like “verb + adjective” in English.

多い is an i-adjective meaning many / numerous.
An i-adjective by itself can end a sentence: 本が多い。 = There are many books.

So the structure is:

図書館には – topic + location (as for the library, in it…)
本が – subject (books…)
多い – predicate adjective (are many)

There’s no separate verb like ある/“to be” needed; 多い already “is” the predicate.
The polite form 多いです simply adds politeness with です, but does not add a new verb meaning.

What is the difference between 多いです and expressions like たくさんあります or たくさんの本があります?

All of these can express “there are many books,” but the grammar is a bit different.

本が多いです。
- 多い is an adjective directly describing 本.
- Literally: Books are many.
本がたくさんあります。
- たくさん is an adverb meaning a lot / in large amount.
- あります is the verb to exist/have for inanimate things.
- Literally: Books exist in large quantity.
たくさんの本があります。
- たくさんの works like many in English, modifying 本 as a noun phrase (many books).
- あります is again to exist/have.
- Literally: There are many books.

In your sentence, using 多いです emphasizes the state/degree (they are numerous).
Using たくさんあります emphasizes existence with quantity (they exist in a large amount).

In everyday speech, both patterns are very common and often interchangeable.

Can I drop です and just say 図書館には本が多い? When is that okay?

Yes, you can say:

図書館には本が多い。

This is grammatically correct, but casual.
Use it:

With friends
In informal writing (notes, diaries, casual online posts)
When you don’t need polite speech

Use 多いです:

In polite conversation
With teachers, strangers, at work (unless you’re very close)
In most textbooks’ example sentences

So:

多い。 → plain / casual
多いです。 → polite / neutral in most situations

Is 図書館 the subject of the sentence, or is 本 the subject?

In:

図書館には本が多いです。

the roles are:

図書館: location phrase, marked with に and topicalized by は. It’s the topic and place, not the subject.
本: marked with が, it is the grammatical subject.
多いです: predicate describing 本.

So structurally:

As for *the library (topic/place), books (subject) are many (predicate).*

This is a common Japanese pattern:

Place + に(は) + Thing + が + Adjective/あります/います
to say “There is/are [thing] in/at [place].”

Could I change the word order, like 本が図書館には多いです? Is that natural?

You can say:

本が図書館には多いです。

It’s grammatically possible and might be used when emphasizing 本 first (“As for the books, in the library there are many…”), but:

The most neutral, natural order for this meaning is the original:
- 図書館には本が多いです。

Japanese word order is somewhat flexible, but topic→location→subject→predicate is very typical and sounds smoother here.
Changing the order can add a nuance of emphasis or contrast, and can sometimes sound a bit marked or less natural in isolation.

How would I say “There aren’t many books in the library” using this pattern?

You can negate 多い to make the negative:

Polite, common:
- 図書館には本が多くないです。
  = There aren’t many books in the library.
Slightly more formal / textbook:
- 図書館には本が多くありません。
Plain (casual):
- 図書館には本が多くない。

Patterns:

多い → 多くない (plain negative)
多くないです / 多くありません (polite negatives)

How do you read the kanji in this sentence, and what are they literally?

Readings:

図書館 → としょかん (toshokan)
- 図 (と) – diagram, plan
- 書 (しょ) – writing, book
- 館 (かん) – building, hall
  Literally: building for writings/diagrams → “library”
本 → ほん (hon) – book
多い → おおい (ooi) – many, numerous

So the whole sentence is read:

としょかんにはほんがおおいです。

Why are there spaces between the words here? Are spaces used in real Japanese writing?

In standard Japanese writing, spaces are not normally used between words.
Your sentence in normal Japanese would be:

図書館には本が多いです。

The version with spaces:

図書館　に　は　本　が　多い　です。

is a teaching aid:

It makes it easier for learners to see the boundaries:
- 図書館 / に / は / 本 / が / 多い / です
It highlights particles like に, は, が.

Once you’re comfortable, you should get used to reading and writing the sentence without spaces.