Learn Hardcore Danish: Regningen er høj i aften, men festen var sjov. - The bill is high tonight, but the party was fun.

You would only add den in front (e.g. den regning) when you want a demonstrative meaning like that bill or this bill, or when the noun has an adjective in front:

den høje regning = the high bill / that high bill

Why is it er høj in the first part and var sjov in the second part? Why change tense?

Er is present tense of at være (to be), and var is past tense.

Regningen er høj i aften = The bill is high tonight (a current situation)
Men festen var sjov = But the party was fun (the party has already happened, you’re talking about it afterwards)

This combination suggests something like: Right now I see that the bill tonight is high, but (looking back) the party was fun.

Other possibilities:

If the party is still going on:
Regningen er høj i aften, men festen er sjov.
“The bill is high tonight, but the party is fun.”
If both things are completely in the past:
Regningen var høj i aftes, men festen var sjov.
“The bill was high last night, but the party was fun.”

Could I say regningen var høj i aften, men festen var sjov with var in both parts?

You can say it, but it sounds odd with i aften, because i aften refers to this coming / current evening, which is normally connected to the present.

If you want both to be clearly in the past, you would normally also push the time expression into the past:

Regningen var høj i aftes, men festen var sjov.
The bill was high last night, but the party was fun.

Keeping i aften and using var in both clauses would confuse the time reference for native speakers.

Why is it høj for the bill? In English we usually say the bill is expensive, not high.

Danish can use høj (high) in contexts where English might use expensive or large:

en høj regning = a big / high bill
huslejen er høj = the rent is high
priserne er høje = the prices are high

So regningen er høj is idiomatic Danish and means the amount on the bill is large.
You could also say:

Regningen er dyr i aften = The bill is expensive tonight

Høj emphasises the amount is high. Dyr emphasises it costs a lot / is expensive. Both are possible, with a slightly different nuance.

What exactly does i aften mean? Is it “in the evening” or “tonight”?

I aften means this evening / tonight (the coming or current evening of today).

Compare:

i aften = this evening / tonight
i morgen aften = tomorrow evening
i går aftes = yesterday evening / last night

So Regningen er høj i aften is “The bill is high tonight”, not a general statement about evenings.

Why is there no article before aften? Why not i den aften?

In time expressions, Danish usually uses bare nouns without articles:

i dag = today
i morgen = tomorrow
i går = yesterday
i aften = this evening / tonight
i weekenden = at / on the weekend (this one takes the definite form)

Using den aften turns it into a specific evening that is already known in the context:

Den aften var regningen høj.
“That evening, the bill was high.”

In your sentence, you’re referring to this evening in a standard time expression, so i aften without an article is correct.

Why is it sjov and not sjovt or sjove after festen var?

Adjectives in Danish agree with the gender and number of the noun, even after er/var.

Basic pattern (singular):

en fest er sjov (common gender: en)
et spil er sjovt (neuter gender: et)

Here:

fest is common gender: en fest
definite form: festen

So the predicate adjective is:

festen var sjov (common gender → sjov)

If it were a neuter noun, you’d use -t:

et show var sjovt = the show was fun

Plural takes -e:

festerne var sjove = the parties were fun

Why is it men festen var sjov, not men var festen sjov?

After men (but), you still have a main clause in Danish, which normally follows V2 word order: the finite verb comes in second position.

In Men festen var sjov:

festen = first element
var = verb, in second position
sjov = the rest of the predicate

So the order men festen var sjov is correct and neutral.

Men var festen sjov is also grammatically possible, but it sounds like a question:
Men var festen sjov? = But was the party fun?

Why is there a comma before men in ..., men festen var sjov?

Men is a coordinating conjunction (like and, or, but in English) that links two main clauses:

Regningen er høj i aften
(Men) festen var sjov

When men connects two independent clauses, Danish usually places a comma before it:

Jeg kom for sent, men de ventede på mig.

If men only links two words or short phrases, you do not put a comma:

stor men dyr = big but expensive
en lille, men vigtig detalje = a small but important detail

In your sentence, two full clauses are being connected, so the comma is correct.

Can i aften move to the beginning of the sentence, like in English: “Tonight the bill is high…”?

Yes. You can say:

I aften er regningen høj, men festen var sjov.

Danish is a V2 language: the finite verb must be in second position. When you move i aften to the front, the verb er must come immediately after it:

I aften (1st position)
er (2nd position – the verb)
regningen høj (rest of the clause)

You cannot say I aften regningen er høj – that would break the V2 rule.

Could you leave out er or var, like Regningen høj i aften?

No. In Danish you normally must include the verb at være (to be) in these sentences:

Regningen er høj i aften. ✓
Festen var sjov. ✓

Leaving out er or var (as you can sometimes drop “is” in casual English, e.g., “You crazy!”) is not standard in Danish. It would sound incomplete or dialectal at best.

Is regningen er høj a common way to say “the bill is expensive,” or is there a more natural phrase?

Regningen er høj is natural and commonly used. It focuses on the high amount on the bill.

Other common ways to complain about the cost:

Regningen er dyr. = The bill is expensive.
Det er dyrt i aften. = It’s expensive tonight.
Det blev en dyr aften. = It turned into an expensive evening.

So your sentence Regningen er høj i aften, men festen var sjov is idiomatic Danish and sounds natural.