Serviceværkstedet reparerer elapparatet.

Because Danish, like German, often forms compound nouns by joining words together.

Serviceværkstedet is built from:

• service
• værksted = workshop / repair shop
• -et = the for a neuter noun

So the base noun is serviceværksted = service workshop / repair shop, and serviceværkstedet means the service workshop.

English usually writes these as separate words more often, but Danish commonly combines them into one word.

The ending -et is the definite article attached to the noun. In English, we put the before the noun; in Danish, it is often added to the end.

So:

• et serviceværksted = a service workshop
• serviceværkstedet = the service workshop

This happens because værksted is a neuter noun in Danish, and many neuter nouns take -et in the definite singular.

Reparerer is the present tense of reparere = to repair.

In Danish, the present tense is often formed by adding -r to the infinitive:

• at reparere = to repair
• reparerer = repairs / is repairing

Unlike English, Danish does not change the verb depending on the subject:

• jeg reparerer = I repair
• du reparerer = you repair
• serviceværkstedet reparerer = the service workshop repairs

So the same present-tense form is used for all persons.

For the same reason: it is a compound noun.

It consists of:

• el = electric / electricity
• apparat = device, appliance
• -et = the

So:

• et elapparat = an electrical device / appliance
• elapparatet = the electrical device / appliance

Danish very often combines this kind of noun into a single word.

El- is a common Danish element related to electricity or electrical things.

Examples:

• elbil = electric car
• elpære = electric light bulb
• elapparat = electrical device / appliance

So in elapparatet, el- tells you that the apparat is electrical.

Because both main nouns are neuter singular nouns in the definite form.

Here are the basic patterns:

• et værksted → værkstedet
• et apparat → apparatet

In this sentence, both nouns are:

1. singular
2. definite
3. neuter

So both take -et.

In this sentence, Danish uses normal subject-verb-object word order:

• Serviceværkstedet = subject
• reparerer = verb
• elapparatet = object

So the pattern is:

The service workshop + repairs + the electrical device

This is similar to standard English word order, which makes this sentence fairly straightforward for English speakers.

Usually, no. In Danish, the definite article is commonly attached to the noun:

• værkstedet = the workshop
• apparatet = the device

You do sometimes see a separate determiner such as den or det, but that is usually when there is an adjective or some special structure:

• værkstedet = the workshop
• det store værksted = the big workshop

So in your sentence, the attached form is the normal one.

It would be:

Et serviceværksted reparerer et elapparat.

That means A service workshop repairs an electrical device.

Compare:

• et serviceværksted = a service workshop
• serviceværkstedet = the service workshop
• et elapparat = an electrical device
• elapparatet = the electrical device

So the sentence you were given uses the definite forms, not the indefinite ones.

It can cover both ideas, depending on context.

Reparerer can mean:

• repairs
• is repairing

Danish present tense often does the job of both the English simple present and present progressive.

So:

• Serviceværkstedet reparerer elapparatet could mean
  The service workshop repairs the electrical device
  or
  The service workshop is repairing the electrical device

The surrounding context usually makes the intended meaning clear.

Yes, æ is important. It is a separate Danish letter, not just a decorated a.

In værksted, the æ has a vowel sound somewhat like the a in cat, though not exactly the same depending on accent and pronunciation style.

A rough guide:

• værksted sounds approximately like VAIRK-sted or VERK-sted, but that is only an approximation for English speakers.
• The r and vowels in Danish often sound less clear-cut than in English, so listening to native audio is very helpful.

The key point is that æ changes the word’s sound and should not be ignored.

Because at reparere is the infinitive form, meaning to repair.

You use:

• at reparere = to repair
• reparerer = repairs / is repairing

A full sentence needs a finite verb, so here Danish uses reparerer, not the infinitive.

Compare:

• Jeg vil reparere elapparatet. = I want to repair the electrical device.
• Jeg reparerer elapparatet. = I repair / am repairing the electrical device.

So in your sentence, reparerer is the correct form.