Peter hinterlässt eine Nachricht.

The infinitive is hinterlassen.

In this sentence, hinterlässt is the 3rd person singular present tense form, used with Peter = he.

A quick pattern:

• ich hinterlasse
• du hinterlässt
• er/sie/es hinterlässt

So Peter hinterlässt matches er hinterlässt.

Because some German verbs change their stem vowel in the 2nd person singular and 3rd person singular in the present tense.

Here, a changes to ä:

• ich hinterlasse
• du hinterlässt
• er hinterlässt

This is a normal pattern for some strong verbs in German.

In this meaning, no: hinterlassen is normally treated as an inseparable verb.

That is why the verb stays together:

• Peter hinterlässt eine Nachricht.

And in the perfect tense:

• Peter hat eine Nachricht hinterlassen.

You do not split it into hinter + lassen here.

Because German main clauses usually follow the V2 rule: the finite verb goes in second position.

Here the sentence starts with the subject:

• Peter = position 1
• hinterlässt = position 2
• eine Nachricht = the rest of the sentence

So this is a very standard German sentence pattern: subject + finite verb + object

Because Nachricht is a feminine noun: die Nachricht.

So with the indefinite article, feminine singular is:

• eine Nachricht

Compare:

• masculine: ein Mann
• neuter: ein Kind
• feminine: eine Nachricht

It is in the accusative case, because it is the direct object of the verb hinterlassen.

The structure is:

• Peter = subject, nominative
• eine Nachricht = direct object, accusative

One detail that can confuse learners: for a feminine singular noun, the indefinite article is eine in both nominative and accusative. So you cannot see the case change from the article alone here.

Because Peter is a proper name, and in standard German proper names usually do not take an article.

So you say:

• Peter hinterlässt eine Nachricht.

not normally:

• Der Peter hinterlässt eine Nachricht.

In some regional varieties of German, people do use articles with names, but that is not the neutral standard form.

Yes, German word order is flexible, but the finite verb still stays in second position in a main clause.

For example:

• Peter hinterlässt eine Nachricht.
• Eine Nachricht hinterlässt Peter.

The second version is grammatically possible, but it sounds more marked and puts emphasis on eine Nachricht. The first version is the most neutral and natural in most situations.

Two very common ways are:

• Peter hinterließ eine Nachricht.
  This is the simple past (Präteritum).

• Peter hat eine Nachricht hinterlassen.
  This is the present perfect (Perfekt).

In everyday spoken German, the Perfekt form is often more common: Peter hat eine Nachricht hinterlassen.