Vi bruger hverken gulvvarmen eller radiatoren.

It means neither ... nor.

In this sentence, hverken introduces the first item, and eller introduces the second:

• hverken gulvvarmen = neither the underfloor heating
• eller radiatoren = nor the radiator

So the pattern is:

• hverken X eller Y = neither X nor Y

This is a very common structure in Danish.

Because hverken ... eller already makes the sentence negative.

So:

• Vi bruger hverken gulvvarmen eller radiatoren. = We use neither the underfloor heating nor the radiator.

You would not normally add ikke here. A sentence like Vi bruger ikke hverken ... would sound wrong in standard Danish.

The -en ending is the usual way Danish marks the definite form of many common-gender nouns, equivalent to English the.

So:

• gulvvarme = underfloor heating
• gulvvarmen = the underfloor heating

and:

• radiator = radiator
• radiatoren = the radiator

Instead of putting a separate word for the in front, Danish often adds the definite ending directly to the noun.

Because Danish very often forms compound nouns as a single word.

Here:

• gulv = floor
• varme = heat/heating
• gulvvarme = underfloor heating

Then the definite ending is added:

• gulvvarmen = the underfloor heating

This is very normal in Danish. English often uses two words where Danish uses one compound word.

Yes, radiator is possible in some contexts, but radiatoren is more natural if you mean a specific radiator known from the situation, or if you are talking about the heating options in a particular room or home.

Compare:

• Vi bruger ikke radiator. = We do not use a radiator / radiator heating.
  This sounds more general or less specific.
• Vi bruger ikke radiatoren. = We do not use the radiator.
  This points to a specific radiator.

In your sentence, both nouns are definite because the speaker is referring to the particular heating systems available in that context.

Bruger is the present tense of at bruge = to use.

So:

• jeg bruger = I use
• vi bruger = we use

In Danish, the present tense can describe:

• something happening now
• a general habit
• a current situation

So this sentence could mean either:

• we are not using them right now, or
• we generally do not use them

The exact meaning depends on context.

Yes. The normal structure here is:

• Vi bruger hverken X eller Y.

That is the most natural order for a main clause with hverken ... eller after the verb.

You can also find hverken earlier in some sentence types, but for a learner, the safest pattern is:

• subject + verb + hverken + item 1 + eller + item 2

So this sentence is a very good model to copy.

Because eller means or, and after hverken, Danish uses eller just like English uses nor after neither.

So the pair is:

• hverken ... eller = neither ... nor

Even though English uses nor, Danish uses eller in this structure.

Yes. It can connect many kinds of sentence elements, not just nouns.

For example:

• Han spiser hverken kød eller fisk.
  He eats neither meat nor fish.

• Jeg kan hverken læse eller skrive kinesisk.
  I can neither read nor write Chinese.

• Det er hverken nemt eller billigt.
  It is neither easy nor cheap.

So in your sentence, it happens to connect two nouns, but the structure is much more flexible.

They are opposites in meaning:

• hverken ... eller = neither ... nor
• enten ... eller = either ... or

Compare:

• Vi bruger hverken gulvvarmen eller radiatoren.
  We use neither the underfloor heating nor the radiator.

• Vi bruger enten gulvvarmen eller radiatoren.
  We use either the underfloor heating or the radiator.

So if you want to express a negative choice between two things, use hverken ... eller.

The main dictionary forms are:

• vi = we
• at bruge = to use
• hverken = neither
• gulvvarme = underfloor heating
• eller = or
• radiator = radiator

In the sentence, two of the nouns appear in their definite singular forms:

• gulvvarmen from gulvvarme
• radiatoren from radiator