Zij ligt op het nieuwe matras te rusten, en haar jas hangt aan de kapstok.

Both zij and ze can mean she.

• zij is the fuller, more stressed form
• ze is the unstressed, more common everyday form

So this sentence could also be:

Ze ligt op het nieuwe matras te rusten, en haar jas hangt aan de kapstok.

Using zij may sound a bit more formal, emphatic, or simply more careful in writing. In normal speech, Dutch speakers often say ze.

Also, zij/ze can mean they, but here it clearly means she because of haar jas = her coat.

Dutch often uses posture verbs where English uses to be.

Here, liggen means to lie or to be lying down. So:

• Zij ligt ... te rusten = She is lying ... resting
• more naturally in English: She is resting while lying on the mattress

Dutch is very natural with verbs like:

• liggen = to lie
• zitten = to sit
• staan = to stand
• hangen = to hang

So instead of just saying someone is somewhere, Dutch often tells you their physical position.

te rusten gives the activity that she is doing while in that position.

So:

• ligt tells you her posture: she is lying down
• te rusten tells you the action: she is resting

Together, liggen te + infinitive is a common Dutch pattern. It means something like:

• to be lying there doing something

Examples:

• Hij ligt te slapen. = He is lying asleep / He is sleeping lying down.
• Ze ligt te lezen. = She is lying reading.

So Zij ligt ... te rusten means she is resting in a lying position.

Yes. Zij is aan het rusten is also correct and means She is resting.

But there is a nuance:

• is aan het rusten focuses on the ongoing action
• ligt te rusten focuses on both the action and the fact that she is lying down

So:

• Zij is aan het rusten. = She is resting.
• Zij ligt te rusten. = She is resting, lying down.

In this sentence, ligt te rusten is more vivid because it matches the mattress.

op means on, and het nieuwe matras means the new mattress.

So:

• op = on
• het matras = the mattress

This is straightforward location language:

• op de tafel = on the table
• op het bed = on the bed
• op het matras = on the mattress

Dutch uses op here just like English uses on.

In standard Dutch, matras is usually a het-word:

• het matras
• een matras

That is why the sentence says het nieuwe matras.

A learner should mainly remember the standard form as het matras. You may occasionally encounter variation in some regions or informal usage, but het matras is the safest and most standard choice.

This is adjective inflection.

In Dutch, adjectives before nouns usually take -e. That is why you get:

• het nieuwe matras
• de grote jas
• een mooie stoel

With het-words, there is one important exception: if the noun is singular, indefinite, and preceded by een, the adjective usually has no -e:

• een nieuw matras
• een groot huis

But with het as in this sentence, you do use -e:

• het nieuwe matras

So the pattern is:

• een nieuw matras
• het nieuwe matras

Here haar is the possessive adjective her.

So:

• haar jas = her coat

This is different from haar as an object pronoun, which can also mean her in sentences like:

• Ik zie haar. = I see her.

So haar can be:

• possessive: her
• object pronoun: her

In this sentence, because it comes before a noun (jas), it is clearly possessive:

• haar jas = her coat

And yes, haar is also the Dutch word for hair, but only as a noun:

• het haar = the hair

So context tells you which meaning it has.

With kapstok (coat rack / coat stand / wall rack), Dutch often uses aan when something is hanging from it:

• De jas hangt aan de kapstok. = The coat is hanging on the coat rack.

This is because the coat is thought of as being attached to or suspended from the rack.

English often says on the coat rack, but Dutch commonly says aan de kapstok.

So this is a good phrase to learn as a set expression:

• aan de kapstok hangen

Just like liggen describes someone lying down, hangen describes something that is hanging.

Dutch likes to be precise about position:

• liggen = lie
• zitten = sit
• staan = stand
• hangen = hang

So for a coat, hangt is the natural choice:

• haar jas hangt aan de kapstok = her coat is hanging on the coat rack

English might simply say her coat is on the coat rack, but Dutch often prefers the more physical verb hangen.

This is normal main-clause word order in Dutch: the finite verb comes in the second position.

Clause 1:

• Zij = position 1
• ligt = position 2

Clause 2:

• haar jas = position 1
• hangt = position 2

So both halves of the sentence follow the usual V2 pattern.

The conjunction en simply connects two main clauses. It does not push the verb to the end. That is why you get:

• Zij ligt ...
• haar jas hangt ...

not a subordinate-clause pattern.

It is just te + infinitive.

The basic verb is:

• rusten = to rest

In this sentence, te rusten works with ligt in the pattern liggen te + infinitive.

So it is not a separable verb like:

• opstaan
• aandoen
• meebrengen

Instead, it is a construction:

• liggen te rusten
• zitten te lezen
• staan te wachten

Yes. In everyday spoken Dutch, many speakers would probably say:

Ze ligt op het nieuwe matras te rusten, en haar jas hangt aan de kapstok.

That sounds very natural.

You might also hear: Ze ligt op het nieuwe matras uit te rusten, en haar jas hangt aan de kapstok.

Here uitrusten means to rest / to recover / to relax after exertion, so it adds a slightly different nuance. But the original sentence is correct as it stands.