Har du din nøgle med, eller ligger den stadig på bordet?

Because Danish forms yes/no questions by putting the finite verb first.

• Statement: Du har din nøgle med. = You have your key with you.
• Question: Har du din nøgle med? = Do you have your key with you?

So Har du ...? is the normal Danish question pattern here.

Here med is part of the expression have noget med, which means to have something with you.

So:

• Har du din nøgle med? = Do you have your key with you?
• Jeg har min telefon med. = I have my phone with me.

Without med, the sentence would be less clearly about physically having the key on you.

Because sin/sit/sine is only used when the possessor refers back to a third-person subject.

• Du har din nøgle med. = You have your key with you.
• Han har sin nøgle med. = He has his own key with him.

Since the subject here is du, Danish uses din, not sin.

Because nøgle is a common-gender noun, an en-word:

• en nøgle

Common-gender singular nouns are normally referred to with den.

• Nøglen → den
• Bogen → den

If it were a neuter noun, an et-word, you would usually use det instead.

Danish often prefers specific position verbs where English often just uses is.

• ligge = lie, be lying
• stå = stand, be standing
• sidde = sit, be situated/fixed in some contexts

A key on a table is naturally thought of as lying, so ligger sounds very natural.

• Den ligger på bordet. = It is lying on the table / It is on the table.

You could sometimes say er på bordet, but ligger is more idiomatic here.

Because the second part continues the question structure after eller.

In Danish, when two yes/no question clauses are joined, each clause can keep question word order:

• Har du din nøgle med, eller ligger den stadig på bordet?

In a normal statement, it would be:

• Den ligger stadig på bordet.

So the inversion in ligger den ... is because this clause is also part of the question.

Stadig means still.

So:

• ligger den stadig på bordet? = is it still on the table?

It suggests that the key was on the table before, and the speaker is asking whether that is still true now.

In this question clause, stadig comes after the subject:

• Ligger den stadig på bordet?

Compare the statement version:

• Den ligger stadig på bordet.

That placement is normal for this kind of adverb in Danish.

Because Danish usually puts the definite article at the end of the noun.

• et bord = a table
• bordet = the table

So:

• på bordet = on the table

English uses a separate word (the), but Danish often uses a suffix instead.

Yes, sometimes, but the meaning is slightly different.

• din nøgle = your key
• nøglen = the key

In context, nøglen might refer to a specific key already known to both speakers. But din nøgle is more explicit and natural if the point is that it is your key.

Because the sentence joins two main clauses:

• Har du din nøgle med
• eller ligger den stadig på bordet

It is standard Danish punctuation to put a comma between coordinated main clauses like this.

A few natural answers would be:

• Ja, jeg har den med. = Yes, I have it with me.
• Ja, jeg har min nøgle med. = Yes, I have my key with me.
• Nej, den ligger stadig på bordet. = No, it’s still on the table.
• Nej, jeg glemte den på bordet. = No, I left it on the table.