Holdkammeraten smiler efter sparen.

Because Danish forms compounds as single words much more consistently than English does.

Here, hold + kammerat becomes holdkammerat, meaning a teammate or team mate. Then the definite ending is added:

holdkammerat = teammate
holdkammeraten = the teammate

So the whole word is built in layers, rather than written as separate words.

The -en is the definite article attached to the noun.

In English, you usually say the teammate with a separate word. In Danish, you usually add the definiteness to the end of the noun:

en holdkammerat = a teammate
holdkammeraten = the teammate

This happens because holdkammerat is a common-gender noun, and common-gender nouns often take -en in the singular definite form.

Because Danish often uses a suffixed definite article instead of a separate word.

So:

the teammate → holdkammeraten
the spar → sparen

A separate word such as den is usually needed when there is an adjective:

holdkammeraten = the teammate
den glade holdkammerat = the happy teammate

So in your sentence, no separate the is needed.

Smiler is the present tense of at smile.

Danish present tense is usually made with -r:

at smile = to smile
smiler = smile / smiles / is smiling, depending on context

Unlike English, Danish does not make a strong everyday distinction between smiles and is smiling in the same way. So smiler can cover both kinds of meaning, depending on the situation.

No. Danish verbs are much simpler here than English verbs.

The present tense form stays the same with all persons:

jeg smiler = I smile
du smiler = you smile
han smiler = he smiles
vi smiler = we smile

So you do not need a special ending for he/she/it beyond the normal present-tense -r.

The important thing to remember is that Danish prepositions do not always match English prepositions word for word.

Efter often means after, but in some expressions or contexts, the best English translation may be different. So if the meaning of the whole sentence has already been given to you, it is better to learn efter here as part of the full Danish expression rather than assuming it will always equal one single English preposition.

This is very common in language learning: prepositions are often one of the least literal parts of a sentence.

For the same reason as holdkammeraten: it is in the singular definite form.

So the pattern is:

en spar = a spar
sparen = the spar

Again, Danish usually marks definiteness on the noun itself instead of putting a separate the in front.

Yes, it is completely normal.

The structure is:

Holdkammeraten = subject
smiler = finite verb
efter sparen = prepositional phrase

So the sentence follows the usual main-clause pattern with the verb in second position. Since the subject comes first, the sentence looks very similar to ordinary English word order here.

Usually no. Holdkammerat is normally used as a general word for a teammate, regardless of sex.

So even though the noun has grammatical gender and takes en, that does not mean it refers only to a man. In Danish, grammatical gender and real-world sex are separate things.

The dictionary forms are:

holdkammerat for holdkammeraten
at smile for smiler
spar for sparen

This is useful because Danish often adds endings directly to words, so when you look them up, you usually need to remove the definite ending or return the verb to its infinitive form.