Learn Hardcore Danish: Kontoret er stille efter arbejdet.

So kontoret er stille literally means “the-office is quiet”. You don’t put a separate “den/det” in front here; -et on kontor already expresses “the”.

What grammatical gender is kontor, and how do I know to use -et?

Kontor is a neuter noun (et‑word): et kontor.
Neuter nouns take -et in the definite singular:

et hus → huset (a house → the house)
et kontor → kontoret (an office → the office)

If it were a common-gender (en‑word) noun, it would end in -en in the definite form, e.g. en bil → bilen (a car → the car).

Why is it stille and not stilt or something with -t?

Stille is one of the adjectives that does not change form; it looks the same for all genders, numbers, and positions.
You say:

et stille kontor (a quiet office)
en stille mand (a quiet man)
kontoret er stille (the office is quiet)

So you never add -t to stille. It already has its “base” form ending in -e.

Is stille an adjective or an adverb in this sentence?

Here stille works as an adjective describing a state of the subject: “the office is quiet.”
In Danish, the pattern subject + er + adjective is very common:

Værelset er rent. – The room is clean.
Kontoret er stille. – The office is quiet.

The same word stille can also be used adverbially in other contexts (e.g. Tal stille – “Speak quietly”), but in this sentence it’s a predicate adjective.

Could I say “kontoret er roligt efter arbejdet” instead of stille? What’s the difference?

You can say kontoret er roligt efter arbejdet, and it is grammatically fine.
Nuance:

stille emphasizes lack of noise (quiet, no sounds).
rolig/roligt emphasizes calmness, lack of activity or stress.

So kontoret er stille focuses more on it being quiet, while kontoret er roligt suggests a calm, relaxed atmosphere (maybe fewer people, less stress, not necessarily completely silent).

Why is it efter arbejdet and not efter arbejde?

Efter arbejdet literally means “after the work” and is the normal idiomatic way to say “after work” as a time-of-day expression.
The definite form arbejdet here refers to the (normal) working period of the day.

Efter arbejde is not the usual way to express “after work” in Danish and will often sound wrong or at least odd in this context. For “after working” you’d more naturally say efter at have arbejdet.

What exactly does arbejdet refer to here – my job in general or today’s workday?

In this sentence, efter arbejdet is understood as “after the working day is over” – after people have finished work and gone home.
It does not normally mean “after my job/career” in a big life sense; that would require a different phrasing (for example, something with når jeg er gået på pension – “when I have retired”).

Why don’t we use a separate word for “the” before arbejdet, like det arbejde?

Just like kontoret, arbejdet already has the definite article attached:

et arbejde = a job / a piece of work
arbejdet = the job / the work

So efter arbejdet = “after the work”.
You would only say det arbejde in special emphasis cases (e.g. det arbejde, du lavede – “that work you did”), not in this simple time expression.

Can I move efter arbejdet to the beginning of the sentence?

Yes: Efter arbejdet er kontoret stille. is perfectly correct and quite natural.
In main clauses, Danish keeps the verb in second position, so when you move the adverbial efter arbejdet to the front, the verb er must still come before the subject:

Kontoret er stille efter arbejdet.
Efter arbejdet er kontoret stille.

How would the sentence change if I wanted to say “The offices are quiet after work” (plural)?

You make kontor plural and definite:

et kontor → flere kontorer → kontorerne

So the sentence becomes:
Kontorerne er stille efter arbejdet.
Here er stille efter arbejdet stays exactly the same; only the subject changes to plural definite kontorerne (“the offices”).

Is efter always followed by a definite noun like arbejdet?

No. Efter is just a preposition meaning after, and it can be followed by many different things:

a definite noun: efter arbejdet, efter festen (after the party)
an indefinite noun or mass noun: efter regn, efter krig (after rain/war – more abstract)
a pronoun or clause: efter det, efter at vi kom hjem (after we came home)

In the specific time expression “after work (today)”, Danish typically uses the definite efter arbejdet.