Varmeregningen er høj.

Because Danish, like German, very often makes compound nouns by joining words together.

• varme = heat / heating
• regning = bill

So varmeregning means heating bill, and varmeregningen means the heating bill.

A useful rule: in Danish compounds, the last part is usually the main noun. Here, regning is the core word, so the whole compound behaves like a kind of regning.

The -en is the definite ending, so it means the.

• en varmeregning = a heating bill
• varmeregningen = the heating bill

Unlike English, Danish often puts the at the end of the noun instead of using a separate word in front.

Because in Danish, the definite article is usually attached to the noun itself.

So English:

• the heating bill

becomes Danish:

• varmeregningen

This is completely normal in Danish. A separate definite word can appear in some other structures, especially when an adjective comes before the noun, for example:

• den høje varmeregning = the high heating bill

But in your sentence, there is no adjective before the noun, so the ending -en is enough.

Because the adjective is agreeing with a singular common-gender noun.

The basic pattern is:

• common gender singular: høj
• neuter singular: højt
• plural: høje

Here, varmeregningen is singular and its head noun regning is a common-gender noun, so the correct form is høj.

Compare:

• Regningen er høj = The bill is high
• Beløbet er højt = The amount is high
• Priserne er høje = The prices are high

The dictionary form gives it away:

• en regning = a bill

If a noun takes en in the indefinite singular, it is usually common gender. If it takes et, it is neuter.

Since varmeregning is built on regning, it follows the same gender:

• en varmeregning
• varmeregningen

Yes. Danish often uses høj in the same kind of way English uses high for numbers, prices, costs, bills, temperatures, and so on.

So Varmeregningen er høj is very natural Danish.

You may also hear dyr in related contexts, but the nuance is a bit different:

• høj regning focuses on the amount being high
• dyr means expensive/costly more generally

For a bill, høj is very common and idiomatic.

Er is the present tense of at være, which means to be.

So:

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

In this sentence, er works just like English is:

• Varmeregningen er høj = The heating bill is high

A rough learner-friendly guide:

• varmeregningen ≈ VAR-muh-rine-ing-en
• høj ≈ hoy

A few warnings:

• Danish pronunciation is much less phonetic than spelling suggests.
• The g in regning is not pronounced like a hard English g.
• høj has the Danish vowel ø, which English does not really have.

If you want to break the long word up while learning, try:

• varme
• regning
• varme-regningen

That usually makes it easier to hear the structure.

Because this is a normal Danish main clause:

• subject: Varmeregningen
• verb: er
• complement: høj

This is the standard pattern in simple statements, just like English:

• The heating bill is high

Danish main clauses usually put the finite verb in second position, and here the subject comes first, so er comes next.

You usually invert the subject and the verb:

• Er varmeregningen høj? = Is the heating bill high?

That is a very common Danish question pattern:

• statement: Varmeregningen er høj
• question: Er varmeregningen høj?

No. Danish does not capitalize all nouns.

It is capitalized here only because it is the first word of the sentence.

So in the middle of a sentence, it would normally be lowercase:

• Jeg synes, at varmeregningen er høj.

This is different from German, where all nouns are capitalized.