Learn Hardcore Russian: Примерно половина этих книг написана иностранными авторами. - Approximately half of these books are written by foreign authors.

Примерно половина этих книг = approximately half of these books (this whole phrase is the subject).
The verb / predicate написана agrees with половина, so половина must be nominative.

If половина were an object (for example, after a verb or a preposition), you could see other cases, like accusative половину, but here it is clearly the subject:
(Что?) Примерно половина этих книг (что сделала?) написана…

Why is it этих книг and not эти книги?

Because after половина (and most words meaning a quantity or part of something), Russian uses the genitive case for the thing being measured.

Structure here:

половина чего? — этих книг (genitive plural: of these books)

So:

половина этих книг = half of these books
эти книги would be nominative plural (these books) and would not correctly show the idea of half of something.

This is the same pattern as:

много книг (a lot of books) — genitive plural
стакан воды (a glass of water) — genitive singular
кусок хлеба (a piece of bread) — genitive singular

Why is написана singular and feminine, not plural написаны?

In Russian, the verb (or short-form passive participle) agrees with the grammatical subject, not with the logical quantity you may be thinking about.

Here the grammatical subject is:

половина — a feminine singular noun.

So the predicate must be:

feminine singular: написана.

If the subject were plural книги, you would use plural:

Эти книги написаны иностранными авторами.
(These books were written by foreign authors.)

But in your sentence, Russian is literally saying:
Half (fem. sg.) of these books is written by foreign authors.
So написана (feminine singular) is the correct agreement.

Is Примерно половина этих книг написаны иностранными авторами (plural) completely wrong?

You do sometimes hear plural agreement in colloquial speech, especially when the speaker is thinking about the actual items (книги) more than about the fraction (половина). So:

Примерно половина этих книг написаны… — can occur in informal conversation.

However, normative, standard Russian prefers agreement with the grammatical subject половина, so:

Примерно половина этих книг написана иностранными авторами. — is the form you should use in careful speech and writing.

For learners, it’s best to stick to the singular: написана.

What exactly is написана grammatically?

Написана is:

a short-form past passive participle,
feminine singular,
from the verb написать (to write in the sense of to complete writing).

Short forms of past passive participles (like написан, написана, написано, написаны) are often used like “passive verbs,” similar in meaning to “is written / was written” in English:

книга написана — the book is written / was written
книги написаны — the books are written / were written

Why is it иностранными авторами? What case is that?

Иностранными авторами is in the instrumental plural.

In Russian, the agent (the “doer”) of a passive construction is expressed with the instrumental case, typically with no preposition:

книги написаны авторами — the books were written by (the) authors
дом построен архитектором — the house was built by the architect

So in your sentence:

написана (кем?) иностранными авторами
(written by whom? — by foreign authors)

English uses the preposition by; Russian uses instrumental case without a preposition.

Why can’t we say написана иностранных авторов instead of иностранными авторами?

Because after a passive participle, the “doer” must be in instrumental, not genitive.

Compare:

Correct: книга написана иностранным автором (instrumental singular)
Incorrect: книга написана иностранного автора (genitive — wrong with this meaning)

Genitive иностранных авторов would need a different structure, for example:

Это книги иностранных авторов.
These are books of foreign authors.

Here, иностранных авторов describes what kind of books they are (genitive), not who wrote them in a passive construction. In your sentence, we are explicitly saying “were written by foreign authors”, so instrumental иностранными авторами is required.

Could this sentence be said in an active voice? How would it look?

Yes, you can express the same idea using active voice. For example:

Иностранные авторы написали примерно половину этих книг.
Foreign authors wrote approximately half of these books.

Here:

иностранные авторы — subject (nominative plural)
написали — active past tense, plural
примерно половину этих книг — direct object (accusative половину, plus genitive этих книг)

The meaning is essentially the same, but:

The original sentence focuses on the books (“Half of these books are written…”).
The active version focuses on the authors (“Foreign authors wrote half of these books.”)

Why is it этих книг and not книг этих? Is книг этих possible?

In practice, этих книг is the normal, neutral word order:

половина этих книг

You can technically say половина книг этих, but:

it sounds unusual and stylistically marked;
it would normally appear only in very specific contexts (poetry, emphasis, or a particular stylistic effect).

Standard, everyday Russian puts the demonstrative before the noun in such noun phrases:

эти книги — these books
этих книг — of these books
та часть этих книг — that part of these books

So for normal speech, use половина этих книг.

What is the role of примерно here, and are there alternatives?

Примерно means approximately / about and modifies the quantity word половина.

примерно половина = about half

Close synonyms in this context:

около половины
приблизительно половина
где-то половина (more colloquial)

They all express approximation, but примерно and приблизительно sound more neutral/formal; около is also very common; где-то is more colloquial.

You can generally place примерно:

before половина: Примерно половина этих книг… (most usual)
or after it, but that’s less natural here (Половина примерно этих книг… sounds odd), so keep it where it is.

Can the word order be changed, for example Иностранными авторами написана примерно половина этих книг?

Yes, Russian word order is quite flexible, and your alternative is grammatically correct:

Иностранными авторами написана примерно половина этих книг.

Differences in nuance:

Original: Примерно половина этих книг написана иностранными авторами.
Focuses first on how many of the books (half).
Alternative: Иностранными авторами написана примерно половина этих книг.
Starts with иностранными авторами, so it puts more emphasis on who the authors are (foreign authors), then tells you that they wrote about half of the books.

Both are correct; choice depends on what you want to highlight.

Why is половина followed by genitive plural этих книг, not genitive singular?

Because we are talking about many books and taking half of the whole group.

Pattern:

When you have “half of X” and X is plural, Russian uses genitive plural:
- половина этих книг — half of these books
- половина студентов — half of the students
- половина домов — half of the houses

If the noun is uncountable or thought of as a mass, you might see genitive singular:

половина воды — half of the water
половина сахара — half of the sugar

So here, книг is a countable plural, so we use genitive plural: этих книг.