Butikken er åpen hele døgnet.

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

• butikk = a shop
• butikken = the shop

So the ending -en is doing the job that the does in English.

Er is the present tense of å være (to be).

So:

• butikken = the shop
• er = is
• åpen = open

This is a very common pattern in Norwegian: subject + er + adjective.

Because the subject butikken is a singular common-gender noun.

Predicate adjectives in Norwegian usually agree with the noun:

• common gender singular: åpen
• neuter singular: åpent
• plural: åpne

So you get:

• Butikken er åpen = the shop is open
• Huset er åpent = the house is open
• Butikkene er åpne = the shops are open

Not exactly.

• åpen = open as a state or condition
• åpnet = opened, from the verb å åpne

So Butikken er åpen means the shop is in an open state.

Butikken er åpnet is much less natural in everyday use if you just mean customers can come in. It sounds more like the shop has been opened or the shop was opened, focusing on the action rather than the state.

Døgn means a full 24-hour period: day and night together.

That is different from dag, which usually means day.

So:

• en dag = a day / daytime / a day as a calendar unit, depending on context
• et døgn = a full 24 hours

This is why hele døgnet is a good way to say around the clock.

Because hele dagen usually means all day, but not necessarily all night.

• hele dagen = all day
• hele døgnet = all 24 hours, including the night

If a shop is open nonstop, hele døgnet is the more accurate expression.

Because Norwegian often uses the definite form in expressions like this when talking about the whole of a time period.

So:

• hele dagen = the whole day
• hele natten = the whole night
• hele uka = the whole week
• hele døgnet = the whole 24-hour period

This sounds natural and idiomatic in Norwegian.

These two vowels are often tricky for English speakers.

• å in åpen is pronounced like the vowel in British English law or more, but shorter and cleaner.
• ø in døgnet has no exact English equivalent. It is similar to the vowel in French deux or German schön.

A rough guide:

• åpen ≈ OH-pen
• døgnet ≈ something like DURG-net or DERG-net, but with rounded lips on the first vowel

The exact sound of ø takes practice, so listening and repeating is very helpful.

Yes. A few common alternatives are:

• Butikken er døgnåpen. = The shop is open 24 hours a day.
• Butikken er åpen døgnet rundt. = The shop is open around the clock.

All three are natural, but Butikken er åpen hele døgnet is very clear and straightforward.