Learn Hardcore Norwegian: Boken tilhører henne, ikke meg.

Because henne is the object form of hun (she). The verb tilhøre (to belong to) takes the possessor as an object: the book belongs to her → boken tilhører henne. Using the subject form (hun) here would be ungrammatical.

Subject: hun
Object: henne

Why is it meg and not jeg in the second part?

In the contrastive tail ikke meg the pronoun stands for the object of the verb (understood as “…does not belong to me”), so you need the object form meg, not the subject form jeg. Compare:

Correct here: Boken tilhører henne, ikke meg.
Different construction: Det er ikke jeg. (after the verb er, prescriptive standard prefers the subject form; in everyday speech you’ll also hear meg)

Could I simply say “Boken er hennes” instead of using tilhører?

Yes. Boken er hennes is very common and a bit more neutral/colloquial. Boken tilhører henne sounds a touch more formal/literary or legalistic. Both mean the same in everyday contexts.

Parallel contrast: Boken er hennes, ikke min.

Why not say “ikke min/mi” after the comma?

You can—if you also change the first half for parallelism:

With object pronouns: Boken tilhører henne, ikke meg.
With possessives: Boken er hennes, ikke min.

Use min or mi according to how you treat the gender of bok (see next questions). With boken you’d typically choose min; with boka, mi is natural.

What’s the difference between boken and boka?

Both are correct Bokmål definite forms of bok (book):

boken = masculine treatment
boka = feminine treatment

Bokmål allows many feminine nouns to be treated as masculine. Stylistically, boka often feels a bit more colloquial/modern; boken can feel a bit more formal/traditional. Match your possessive accordingly:

boken min / boka mi
Boken er min / Boka er mi

How do henne and hennes differ?

henne = object pronoun (her): Boken tilhører henne.
hennes = possessive (her/hers): Boken er hennes.

Note: The reflexive sin/si/sitt/sine cannot be used predicatively, so you cannot say Boken er sin. Use hennes in predicative position.

What’s the difference between tilhøre and høre til?

tilhøre is the standard verb for ownership: Boken tilhører henne.
høre til often means “belong to” in the sense of being part of a group/category/place: Dette kapitlet hører til del 2. Some speakers do use høre til for possession, but tilhøre or er hennes is safer and more idiomatic for ownership.

Could I use eie (to own) here?

You’d normally flip the subject if you use eie:

Hun eier boken. (She owns the book.)

eie foregrounds legal ownership; tilhøre or er hennes foregrounds belonging/possession from the item’s perspective. In many contexts they overlap.

Is the word order with ikke correct? Could I say “Boken ikke tilhører meg”?

The correct main-clause order is verb-second, with ikke after the finite verb:

Correct: Boken tilhører ikke meg.
Incorrect: Boken ikke tilhører meg.

In a subordinate clause, ikke comes before the verb: at boken ikke tilhører meg.

Do I need the comma before ikke meg?

It’s standard and helpful here because ikke meg is an elliptical contrast (“…not me”). You could also write:

Boken tilhører henne, men ikke meg. Without the comma, Boken tilhører henne ikke meg looks odd and is best avoided.

Can I say “ikke til meg” instead of “ikke meg”?

Yes, Boken tilhører henne, ikke til meg is possible and emphasizes the preposition til (“to me”) explicitly. However, the shorter ikke meg is more idiomatic in this contrastive tag.

How would I ask “Whose book is it?” using this vocabulary?

Several natural options:

Hvem tilhører boken/boka? (formal-ish)
Hvem eier boken/boka?
Very common in speech: Hvem sin bok er dette? (or denne) More formal/written: Hvis bok er dette? (“whose”), though hvem sin is what you’ll hear most.

Why is it not “Det er ikke meg” here?

Different structure. With the copula er and a pronoun complement, prescriptive Bokmål prefers the subject form:

Det er ikke jeg (prescriptive)
Det er ikke meg (very common in speech) Our sentence isn’t a copular clause; it has tilhører with an object, so meg is required.

Pronunciation tips for the sentence?

Approximate guide (Standard East Norwegian):

Boken: BOO-ken (long oo; final -en is unstressed)
tilhører: TIL-hø-rer (ø like French eu in “deux”; r is a tap/trill depending on dialect)
henne: HEN-ne (double n → short vowel)
ikke: IK-ke (double k → short vowel, long consonant)
meg: like English “my” (regional variants exist) Stress falls on the first syllable of tilhører and henne.

Can I replace boken with a pronoun?

Yes, for a previously mentioned common-gender noun like bok, use den:

Den tilhører henne, ikke meg.