Wir tragen den Teamnamen auf dem Antwortbogen ein.

Because eintragen is a separable verb.

• The full infinitive is eintragen = to enter / to fill in / to write in
• In a normal present-tense main clause, the prefix ein- separates and moves to the end:
  • Wir tragen ... ein.
• In the infinitive or with a modal verb, it stays together:
  • Wir müssen den Teamnamen eintragen.
  • Den Teamnamen einzutragen ist wichtig.

So tragen is the conjugated part, and ein goes to the end.

Not by itself in this sentence. On its own, tragen often means to carry, to wear, or to bear.

But here the verb is not just tragen. It is eintragen, which has a different meaning: to enter, to fill in, to write into a form/list/document.

So you should understand the sentence as using the full verb eintragen, not the basic verb tragen alone.

Because den Teamnamen is the direct object of the verb eintragen, so it is in the accusative case.

• Nominative: der Teamname
• Accusative: den Teamnamen

Here is the pattern:

• Wer trägt ein? — wir
• Was tragen wir ein? — den Teamnamen

That is why German uses den Teamnamen.

Also note that Teamname is a masculine noun, so:

• der Teamname
• den Teamnamen

Because Teamname is one of the masculine nouns that add -n in the accusative singular.

So you get:

• Nominative: der Teamname
• Accusative: den Teamnamen
• Dative: dem Teamnamen
• Genitive: des Teamnamens

This happens because the noun ends in -e in the basic form. Many masculine nouns like that add -n in some cases.

Because auf can take either dative or accusative, depending on the meaning.

• Dative = location, where something is
• Accusative = direction, where something is moving to

In this sentence, auf dem Antwortbogen means something like on the answer sheet as a location: the name is being written in that place / on that sheet.

So German uses the dative:

• auf dem Antwortbogen

If you were emphasizing movement toward a surface, German might use accusative in other contexts, but with writing on a document, dative is very natural.

It is dative singular.

The noun is:

• der Antwortbogen = the answer sheet

After auf, German uses:

• dative for location
• accusative for direction

Since the phrase tells us where the team name is entered, it uses dative:

• auf dem Antwortbogen

Forms:

• Nominative: der Antwortbogen
• Accusative: den Antwortbogen
• Dative: dem Antwortbogen

Yes, the sentence can be rearranged, but the conjugated verb still stays in second position, and the separable prefix ein still goes to the end in a main clause.

For example:

• Wir tragen den Teamnamen auf dem Antwortbogen ein.
• Den Teamnamen tragen wir auf dem Antwortbogen ein.
• Auf dem Antwortbogen tragen wir den Teamnamen ein.

These all mean basically the same thing, but the emphasis changes.

What does not change:

• tragen stays in second position
• ein stays at the end

Because German often forms compound nouns by joining words together into one word.

Here:

• Team
  • Name = Teamname
• In the sentence, the accusative form is Teamnamen

This is very normal in German. English often uses separate words where German prefers one compound noun.

Other examples:

• Antwortbogen = Antwort
  • Bogen
• Haustür
• Schulbus

Antwortbogen is a compound noun:

• Antwort = answer
• Bogen = sheet / form

So Antwortbogen means something like:

• answer sheet
• response sheet
• sometimes answer form, depending on context

It is masculine:

• der Antwortbogen

Yes, absolutely.

• den Teamnamen = the team name
• unseren Teamnamen = our team name

Both are grammatical. The difference is just how specific you want to be.

Examples:

• Wir tragen den Teamnamen auf dem Antwortbogen ein.
• Wir tragen unseren Teamnamen auf dem Antwortbogen ein.

The second one sounds a bit more explicit because it clearly says the team name belongs to us.

Yes. Wir tragen ... ein is present tense.

But in German, the present tense is often used for both:

• what is happening now
• what will happen soon / next

So depending on context, it can mean:

• We are entering the team name on the answer sheet.
• We enter / write in the team name on the answer sheet.
• We will enter the team name on the answer sheet.

German uses the present tense very broadly, more often than English does.