Koppen er populær i hjemmet.

In Danish, the is often built into the noun itself instead of being a separate word.

• kop = cup
• koppen = the cup
• hjem = home
• hjemmet = the home

So in Koppen er populær i hjemmet, both koppen and hjemmet are definite forms.

A useful extra point: when there is an adjective directly before the noun, Danish often uses both a separate word and the ending:

• den populære kop = the popular cup

But in your sentence, populær comes after er, so you just get koppen.

Because these nouns belong to different grammatical genders in Danish.

• en kop → koppen
• et hjem → hjemmet

Danish has two genders:

• common gender: usually takes en in the indefinite form and -en in the definite form
• neuter: usually takes et in the indefinite form and -et in the definite form

So:

• kop is a common-gender noun: en kop / koppen
• hjem is a neuter noun: et hjem / hjemmet

Er is the present tense of at være = to be.

So:

• jeg er = I am
• du er = you are
• han/hun/den/det er = he/she/it is
• vi er = we are

Danish does not change this present-tense form for different persons the way English does with am / is / are. It is simply er for all of them.

Because it is a predicate adjective: it comes after er and describes the subject.

In Danish, predicate adjectives agree with the noun in number, and sometimes gender:

• Koppen er populær.
  common gender singular

• Hjemmet er populært.
  neuter singular

• Kopperne er populære.
  plural

So in your sentence, koppen is singular common gender, so the basic form populær is used.

These are related, but they are not exactly the same.

• hjemme usually means at home
• i hjemmet literally means in the home or in the household

So i hjemmet can sound a bit more formal, literal, or general. It often refers to the home as an environment or setting.

Compare:

• Koppen er populær hjemme.
  The cup is popular at home.

• Koppen er populær i hjemmet.
  The cup is popular in the home / in the household.

In many everyday situations, English speakers may expect hjemme, but i hjemmet is perfectly grammatical and can sound slightly more formal or descriptive.

It can mean either, depending on context.

Hjem is closer to home than house. That means it can refer to:

• the place where someone lives
• the home environment
• the household or family setting

If you specifically mean the building, huset = the house may be more concrete.

So:

• i hjemmet = in the home / in the household
• i huset = in the house

The basic order here is:

Subject + verb + adjective + prepositional phrase

So:

• Koppen = subject
• er = verb
• populær = predicate adjective
• i hjemmet = prepositional phrase

This is a very normal Danish sentence pattern.

Danish is also a verb-second language, which means the finite verb usually comes in the second position in main clauses. For example, if you move i hjemmet to the front, the verb still stays second:

• I hjemmet er koppen populær.

That word order is also correct.

Yes, it is grammatically correct and understandable.

That said, whether it sounds natural depends on the situation. If you simply mean that people at home like the cup, Danish speakers might also say something like:

• Koppen er populær derhjemme.
• Koppen er populær hjemme hos os.

But Koppen er populær i hjemmet is still a valid sentence, especially if you want a slightly more formal or general tone.

Yes. Populær does not have to describe only people.

It can describe anything that is well-liked, favored, or widely used:

• Han er populær. = He is popular.
• Sangen er populær. = The song is popular.
• Koppen er populær. = The cup is popular.

So using populær with an object is completely normal if the context makes sense.

You would usually change the nouns like this:

• koppen → en kop
• hjemmet → et hjem

So a possible indefinite version would be:

• En kop er populær i et hjem.

However, that changes the meaning. It no longer refers to a specific cup or a specific home. It becomes more general, like a cup is popular in a home, which is less likely in ordinary conversation unless you are speaking very generally.