Småkagen ligger på tallerkenen ved siden af koppen.

Because they are in the definite singular form: the cookie, the plate, the cup.

In Danish, a noun often gets the as a suffix instead of using a separate word:

• en småkage = a cookie

• småkagen = the cookie

• en tallerken = a plate

• tallerkenen = the plate

• en kop = a cup
• koppen = the cup

This is one of the most important patterns in Danish.

In Danish, the definite article is usually attached to the end of the noun.

So instead of saying something like the cookie, Danish often says:

• småkagen = the cookie

rather than a separate article before the noun.

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

• småkagen = the cookie
• den store småkage = the big cookie

So in your sentence, no separate word for the is needed.

Danish often uses specific position verbs where English would simply use is.

Here, ligger comes from at ligge, which means to lie or to be lying. It is used for things that are resting horizontally or simply located somewhere.

So:

• Småkagen ligger på tallerkenen = The cookie is lying/is on the plate

Using er is not always wrong in every context, but ligger sounds more natural here because it describes the cookie’s physical position.

Danish commonly does this with objects:

• bogen ligger på bordet = the book is on the table
• koppen står på bordet = the cup is on the table

Notice that Danish may choose different verbs depending on how something is positioned.

Ved siden af means next to or beside.

It is a fixed expression:

• ved = by / at
• siden = the side
• af = of

So literally it is something like at the side of, but in normal English you should think of it simply as next to.

Example:

• ved siden af koppen = next to the cup

Because a cookie is normally understood as being on a plate, not in it.

So Danish uses:

• på tallerkenen = on the plate

Just like English, Danish prepositions often follow the physical image:

• på = on
• i = in

A plate is usually treated as a surface, so på is the natural choice.

Yes, småkage is one word. Danish forms compound nouns very often.

It is made from:

• små = small
• kage = cake

So literally it is something like small cake, but the actual meaning is cookie or sometimes biscuit, depending on context.

This is very normal in Danish: compounds are usually written as one word, not two.

The indefinite forms are:

• en småkage = a cookie
• en tallerken = a plate
• en kop = a cup

All three are common gender nouns, which is why they use en rather than et.

Then the definite forms are:

• småkagen
• tallerkenen
• koppen

So a useful pattern is:

• en kop → koppen
• en tallerken → tallerkenen
• en småkage → småkagen

No. In ved siden af koppen, the word koppen is not in a special case form.

Danish has very little case marking on nouns. Here, koppen is simply the ordinary definite form of kop.

So:

• af is just the preposition of
• koppen still means the cup

Unlike German, Danish does not usually change noun endings for different cases. The main exception is that pronouns can change form, but ordinary nouns usually do not.

Most naturally, it describes the plate.

So the sentence is usually understood as:

• The cookie is on the plate that is next to the cup.

That is because ved siden af koppen comes right after tallerkenen, so it most naturally attaches to that noun phrase.

If you wanted to make it clearer that the cookie is next to the cup, you would probably rephrase the sentence.

So yes, there can be some ambiguity, but the most natural reading is that the plate is next to the cup.

This sentence has the normal Danish main-clause order:

• Småkagen = subject
• ligger = verb
• på tallerkenen ved siden af koppen = adverbial/prepositional phrase

So it follows the common pattern:

• subject + verb + other information

But Danish is a V2 language, which means the finite verb usually comes in the second position in main clauses.

For example, you can move the location phrase to the front:

• På tallerkenen ved siden af koppen ligger småkagen.

That still means the same thing, but now the location is emphasized. Notice that ligger still stays in second position.

Yes, that would be understandable, and in some situations it may sound fine.

But ligger is more idiomatic because Danish often prefers a position verb for the location of physical objects.

So:

• Småkagen er på tallerkenen = understandable
• Småkagen ligger på tallerkenen = more natural and descriptive

A good habit in Danish is to notice when speakers use:

• ligge for things lying
• stå for things standing
• sidde for things sitting / attached / positioned in a certain way

This is a big difference from English.