Jeg bemærkede ikke, at døren stod åben hele tiden.

Because the speaker is talking about something they did not notice at an earlier time.

Bemærkede is the past tense of bemærke.

• jeg bemærker = I notice / am noticing
• jeg bemærkede = I noticed

So Jeg bemærkede ikke ... means I didn’t notice ...

In a normal Danish main clause, the finite verb usually comes in second position. This is often called the V2 rule.

So in:

Jeg bemærkede ikke ...

the order is:

• Jeg = subject
• bemærkede = finite verb
• ikke = negation

That is the normal Danish word order for a main clause.

Compare:

• Jeg så det ikke. = I didn’t see it.
• Hun forstod det ikke. = She didn’t understand it.

Here at means that and introduces a subordinate clause.

So:

• Jeg bemærkede ikke = I didn’t notice
• at døren stod åben hele tiden = that the door was standing open the whole time

In English, that is often optional, but in Danish at is very common and usually included in this kind of sentence.

Because at døren stod åben hele tiden is a subordinate clause, and Danish often marks that with a comma.

So the comma separates:

• the main clause: Jeg bemærkede ikke
• the subordinate clause: at døren stod åben hele tiden

You will very often see a comma before at in written Danish.

Both can be possible, but stod åben is very natural in Danish when talking about a door, window, gate, etc.

Stå is often used for something being in a certain physical position or state, especially when it is left that way.

So:

• Døren stod åben = The door stood open / was left open
• Døren var åben = The door was open

Stod åben can sound a bit more concrete and physical, as if the door was in that open state during that time.

English often just says was open, but Danish frequently uses stå here.

Because stå is an irregular verb.

Its main forms are:

• at stå = to stand
• står = stands / is standing
• stod = stood
• har stået = has stood

So døren stod åben uses the correct past form of stå.

Because døren is a common-gender singular noun, and predicate adjectives agree with that.

• en dør = a door
• døren = the door

With a common-gender singular noun, the adjective is usually in its basic form:

• Døren er åben.
• Døren stod åben.

With a neuter noun, you would usually get -t:

• Vinduet er åbent. = The window is open.

With plural, you would usually get -e:

• Dørene er åbne. = The doors are open.

Here hele tiden means something like the whole time or all along.

In this sentence, it means the door was open during the entire relevant period, and the speaker failed to notice that.

So the sense is:

• not just sometimes
• not for a moment
• but throughout the period being talked about

Depending on context, hele tiden can also mean all the time, but here the whole time is the most natural understanding.

Yes, Danish adverbials can often move, although the version you have is very natural.

The most neutral version is:

Jeg bemærkede ikke, at døren stod åben hele tiden.

You might also see other placements for emphasis, but the end position is very common because hele tiden modifies the whole situation in the subordinate clause.

They are close, but not always identical in tone.

• bemærke is often slightly more formal or neutral: to notice
• lægge mærke til is very common in everyday Danish and also means to notice

So these are both possible:

• Jeg bemærkede ikke, at døren stod åben hele tiden.
• Jeg lagde ikke mærke til, at døren stod åben hele tiden.

The second one may sound a bit more conversational.

Because in the clause introduced by at, Danish uses subordinate-clause word order.

Here the subject comes before the verb:

• at døren stod åben hele tiden

That is normal after at.

A useful comparison:

• Main clause: Døren stod åben.
• Subordinate clause: ... at døren stod åben.

If you added ikke inside the subordinate clause, it would usually come before the finite verb:

• ... at døren ikke stod åben

That difference is one of the key things learners notice about Danish clause structure.

In standard Danish, døren is pronounced roughly like DUR-en or DUR’n, but that is only an approximation.

The tricky part is the soft d in døren. In many Danish words, d between vowels or after certain sounds is not pronounced like a hard English d. It becomes a much softer sound.

So døren does not sound like a clear English door-en.

Also:

• ø is not an English vowel, so learners usually need practice with it
• the ending -en is often reduced in natural speech

A rough learner-friendly approximation is better than a perfect one at first, but this word is definitely worth listening to from native audio.

Yes, that is a very literal translation, and it helps show the Danish structure.

But in natural English, you would usually say:

• I didn’t notice that the door was open the whole time.

So the Danish uses stod åben, while idiomatic English usually prefers was open. That is a very common difference between the two languages.