Vi bruger det fælles køkken på kontoret, men alle rydder op efter sig selv.

Because køkken is a neuter noun in Danish: et køkken.

When Danish has a definite noun with an adjective, it normally uses:

det/den + adjective + noun

So:

• et køkken = a kitchen
• køkkenet = the kitchen
• det fælles køkken = the shared kitchen

It is det, not den, because køkken is neuter.

Without the adjective, you would simply say køkkenet.

Because fælles is one of those adjectives that usually stays the same in all forms.

You can compare:

• en fælles plan
• et fælles køkken
• de fælles regler

So unlike many adjectives, it does not add extra endings here.

In Danish, på is often used for workplaces and institutions, where English would often say at:

• på kontoret = at the office
• på skolen = at the school
• på hospitalet = at the hospital

i kontoret would sound more like inside the office room physically, and is less natural here.

So på kontoret is the normal choice for at the office.

Yes.

kontor is the basic noun, and because it is a neuter noun (et kontor), the definite ending is -et:

• et kontor = an office
• kontoret = the office

The same pattern appears with køkken:

• et køkken = a kitchen
• køkkenet = the kitchen

Here it means everyone or all the people involved.

In English, everyone sounds more natural in this sentence because the idea is that each person does their own part:

• alle rydder op efter sig selv = everyone cleans up after themselves

So although alle can mean all, here everyone is usually the best match.

It is a very common Danish verb + particle combination.

The full expression is at rydde op, which means:

• to tidy up
• to clean up
• to clear up

So:

• rydder = present tense of rydde
• op = the particle that gives the meaning up

Examples:

• Jeg rydder op. = I’m tidying up.
• Vi skal rydde op. = We have to tidy up.

It is an idiomatic expression meaning:

• after oneself
• after themselves

So rydde op efter sig selv means to clean up after oneself.

Here, efter does not mean time-related after. It is part of the fixed expression.

Because sig is the normal third-person reflexive pronoun in Danish when it refers back to the subject.

That is true even when the subject is plural in meaning, as here with alle.

So:

• alle rydder op efter sig selv = everyone cleans up after themselves

The word selv adds emphasis, similar to themselves in English.

A useful contrast:

• De vasker sig. = They wash themselves.
• De vasker dem. = They wash them. (some other people)

So sig shows that the action goes back to the subject.

Because men joins two main clauses, and after it, Danish starts a new clause with normal main-clause word order:

subject + verb

So:

• Vi bruger det fælles køkken på kontoret
• men alle rydder op efter sig selv

Both parts are independent main clauses.

You get inversion in Danish when some other element comes first in the clause, for example:

• På kontoret bruger vi det fælles køkken.

There, på kontoret is first, so the verb comes before the subject: bruger vi.

But after men, that does not happen automatically.

Because Danish verbs do not change for person or number in the present tense.

So you get:

• jeg bruger
• du bruger
• vi bruger
• de bruger

And likewise:

• jeg rydder
• vi rydder
• alle rydder

This is much simpler than English in some ways, because Danish does not have different present-tense forms like I use vs he uses.

Yes — two chunks are especially useful:

• det fælles køkken = the shared kitchen
• rydde op efter sig selv = clean up after oneself

These are natural, everyday combinations, and learning them as whole pieces will help you sound more idiomatic than translating word by word.