Die Mitarbeiterin sieht mein Armband und lässt mich durch.

Because Mitarbeiterin is a feminine singular noun, so in the nominative case it takes the article die.

• der Mitarbeiter = the male employee
• die Mitarbeiterin = the female employee

Here Die Mitarbeiterin is the subject of the sentence, so nominative is used.

Yes. The ending -in usually makes a noun explicitly feminine when talking about a person.

• Mitarbeiter = male employee, or sometimes employee in a general sense depending on context
• Mitarbeiterin = female employee

So this sentence clearly tells you the staff member is female.

Because all German nouns are capitalized, not just proper names.

So in this sentence:

• Mitarbeiterin = noun
• Armband = noun

Both must begin with a capital letter.

Because the subject is Die Mitarbeiterin, which is she in English, so the verb must be in the 3rd person singular.

The infinitive is sehen = to see.

Present tense:

• ich sehe
• du siehst
• er/sie/es sieht

So Die Mitarbeiterin sieht = The employee sees.

Also note that sehen is a strong verb, so its stem vowel changes from e to ie in du and er/sie/es forms.

Because Armband is a neuter noun: das Armband.

After a possessive word like mein, a neuter singular noun in the accusative has no extra ending here:

• nominative: mein Armband
• accusative: mein Armband

So mein Armband is correct.

Compare that with a masculine noun:

• mein Pass = nominative
• meinen Pass = accusative

That is why you do not see meinen here.

You usually learn a German noun together with its article:

• das Armband

Also, in compound nouns, the last part determines the gender. Here the last part is Band, and das Band is neuter, so das Armband is also neuter.

It is in the accusative case because it is the direct object of sieht.

Ask the question:

• The employee sees what?
• mein Armband

That makes mein Armband the thing being seen, so it is the direct object.

Because mich is the accusative form of ich.

• ich = I
• mich = me

In this sentence, the employee is doing the action, and I/me am the person being let through. So German uses the object form:

• lässt mich durch = lets me through

Not:

• lässt ich durch ❌

It comes from the separable verb durchlassen, which means to let through or to allow through.

So:

• Sie lässt mich durch. = She lets me through.
• infinitive: durchlassen

The verb lassen is conjugated here as lässt, and durch is the separated prefix.

Because durchlassen is a separable verb.

In a normal main clause, German splits separable verbs:

• the conjugated part goes in the usual verb position
• the prefix goes to the end

So:

• durchlassen → Sie lässt mich durch.

You can compare:

• Ich stehe auf. from aufstehen
• Er kommt mit. from mitkommen

But when the verb is not split, it stays together:

• Sie will mich durchlassen.
• ..., weil sie mich durchlässt.

Because the subject is the same in both parts of the sentence: Die Mitarbeiterin.

German can omit the repeated subject after und when it is clear:

• Die Mitarbeiterin sieht mein Armband und lässt mich durch.

This is understood as:

• Die Mitarbeiterin sieht mein Armband und sie lässt mich durch.

The version with sie is possible, but leaving it out is very natural and common.

Yes. It contains two coordinated actions joined by und:

1. Die Mitarbeiterin sieht mein Armband
2. (Die Mitarbeiterin) lässt mich durch

So the meaning is: first she sees the wristband, and as a result she lets me pass.

Yes, very often. Jemanden durchlassen is commonly used when someone is allowed to pass a barrier, gate, entrance, or security point.

It fits situations like:

• a concert entrance
• a staff-only area
• airport or event security
• a door or gate

So with Armband in the sentence, it sounds very natural: the wristband shows authorization, and the employee lets the speaker through.