Am Eingang umarmt meine Schwester zuerst unsere Mutter.

Because German often puts a time, place, or other adverbial phrase at the beginning of the sentence for emphasis or style.

Here, Am Eingang means at the entrance and sets the scene first.

A more neutral word order would be:

Meine Schwester umarmt am Eingang zuerst unsere Mutter.

Both are correct. Starting with Am Eingang simply gives that location a little more prominence.

This is because German follows the verb-second (V2) rule in main clauses.

That means the finite verb must be in the second position, not necessarily right after the subject.

In this sentence:

• Am Eingang = position 1
• umarmt = position 2
• meine Schwester = comes after the verb

So:

• Am Eingang umarmt meine Schwester zuerst unsere Mutter.

If the subject comes first, then the verb still stays second:

• Meine Schwester umarmt am Eingang zuerst unsere Mutter.

Both are grammatical.

Am is a contraction of:

• an dem → am

So:

• am Eingang = an dem Eingang

In English, this is usually translated as at the entrance.

This is very common in German:

• im = in dem
• am = an dem
• vom = von dem
• zum = zu dem

Because this sentence describes a location, not movement toward a destination.

German prepositions like an can take either:

• dative for location
• accusative for movement toward something

Here, the hugging happens at the entrance, so it is static location:

• am Eingang = dative

Compare:

• Am Eingang wartet sie. = She is waiting at the entrance.
• Sie geht an den Eingang. = She goes to the entrance.

Meine Schwester is the subject, so it is in the nominative case.

That is why the possessive is meine:

• meine Schwester = my sister

If it were in a different case, the form might change. For example:

• Ich sehe meine Schwester. = I see my sister.
  Here, it is accusative, but for feminine singular, meine stays the same.

So in this sentence, meine Schwester is the person doing the action.

Unsere Mutter is the direct object, so it is in the accusative case.

The reason it does not look obviously different is that for feminine singular, the forms of the possessive often look the same in nominative and accusative:

• nominative: unsere Mutter
• accusative: unsere Mutter

So even though the case is accusative, the words do not change here.

You can tell it is the object because of the sentence structure and meaning:

• meine Schwester = the one hugging
• unsere Mutter = the one being hugged

Because possessive words like mein, dein, unser, ihr usually take the place of an article.

So German normally says:

• meine Schwester
• unsere Mutter

not:

• die meine Schwester
• die unsere Mutter

This works much like English my sister or our mother, where you also do not add the.

Zuerst means first.

It usually means that this action happens before another action or before another person is hugged.

So the sentence suggests a sequence, such as:

• first she hugs our mother
• then maybe someone else afterward

It can sometimes also feel like first of all, depending on context.

German allows some flexibility, although some positions sound more natural than others.

Your sentence has:

• Am Eingang umarmt meine Schwester zuerst unsere Mutter.

You could also say:

• Meine Schwester umarmt am Eingang zuerst unsere Mutter.
• Zuerst umarmt meine Schwester am Eingang unsere Mutter.

These versions are grammatical, but the emphasis changes:

• zuerst in the middle sounds neutral
• Zuerst at the beginning gives stronger emphasis to first

No. Umarmen is not separable.

That is important because some German verbs beginning with prefixes do separate, but umarmen does not.

So you say:

• Sie umarmt ihre Mutter.

and not something like:

• Sie armt ihre Mutter um.

That would be wrong.

It is present tense.

The verb is umarmen and the er/sie/es form is:

• er/sie/es umarmt

Since meine Schwester is third-person singular, the verb is umarmt.

So the structure is:

• meine Schwester umarmt = my sister hugs / is hugging

Depending on context, German present tense can correspond to English:

• hugs
• is hugging

Yes, depending on tone and context.

Unsere Mutter is literally our mother, but in natural English it could also be:

• our mom
• our mum (in British English)

The German sentence itself is neutral. Whether you choose mother or mom/mum in English depends on style, not on a grammatical difference in the German.