Drinken er kold.

Because Danish usually adds the definite article to the end of the noun.

• en drink = a drink
• drinken = the drink

So instead of using a separate word like English the, Danish often uses a suffix such as -en or -et.

This is a very good question, because both forms exist.

• en drink / drinken usually means a mixed drink or cocktail
• en drik / drikken is more general and can mean a beverage/drink

So Drinken er kold is fine if the noun is really drink. If you mean the beverage in a more general sense, learners will also often see Drikken er kold.

Because drink is a common-gender noun in Danish, an en-word: en drink.

With a singular common-gender noun, the adjective after er normally stays in its basic form:

• Drinken er kold.

If the noun were a neuter et-word, you would usually get -t on the adjective:

• Glasset er koldt. = The glass is cold.

So kold matches en drink.

Normally, no. Drinken already means the drink, so adding den would usually be wrong here.

In Danish, den/det/de is often used when there is an adjective directly before the noun:

• drinken = the drink
• den kolde drink = the cold drink

But in Drinken er kold, the adjective comes after er, so you do not use den.

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

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

• jeg er = I am
• du er = you are
• han er = he is
• vi er = we are

So er works for all persons in the present tense.

Yes, in this simple sentence it is very similar.

The pattern is:

• Drinken = subject
• er = verb
• kold = adjective/complement

So the structure is basically subject + verb + adjective, just like The drink is cold.

You switch the verb and the subject:

• Er drinken kold? = Is the drink cold?

This kind of inversion is very common in Danish.

Yes. Kolde is used with plurals, and also in some other adjective patterns.

For example:

• Drinkene er kolde. = The drinks are cold.
• de kolde drinks = the cold drinks

But with one singular en-word noun and an adjective after er, kold is the correct form:

• Drinken er kold.