Jeg ville have en kop kaffe, hvis jeg ikke skulle op tidligt i morgen.

Because ville is the form Danish normally uses for a hypothetical or unreal situation. Here it works like English would.

So Jeg ville have en kop kaffe means something like I would have a cup of coffee under certain circumstances.

If you said Jeg vil have en kop kaffe, that would sound much more direct, like I want a cup of coffee or sometimes I’ll have a cup of coffee.

After a modal verb like ville, the next verb stays in the infinitive, without at.

So:

• ville have = would have
• not ville har

This is the same pattern you see with other modal verbs:

• jeg kan komme = I can come
• jeg skal gå = I have to go
• jeg ville have = I would have

The finite verb in the clause is ville, so have has to remain non-finite.

Because skulle here is not being used as a normal past tense referring to past time. It is being used to mark a hypothetical condition.

This is very similar to English, where we often use a past form for unreal situations:

• If I didn’t have to get up early tomorrow...

The time is future, but the grammar is remote/hypothetical.

So:

• Hvis jeg ikke skal op tidligt i morgen = a real/open possibility
• Hvis jeg ikke skulle op tidligt i morgen = a more hypothetical or unreal situation

Yes, in a way. Skulle op is a very common shortened way of saying skulle stå op, meaning have to get up.

So this part means:

• skulle op tidligt = have to get up early

Danish often drops stå in this expression when the meaning is obvious:

• Jeg skal tidligt op = I have to get up early
• Vi måtte tidligt op = We had to get up early

So op is the particle from stå op.

Because this is a subordinate clause, and Danish subordinate clauses have different word order from main clauses.

After hvis, the usual order is:

• subject + sentence adverb + finite verb

So:

• hvis jeg ikke skulle op ...

not:

• hvis jeg skulle ikke op ...

Compare:

• main clause: Jeg skulle ikke op tidligt
• subordinate clause: hvis jeg ikke skulle op tidligt

The word ikke comes before the finite verb in subordinate clauses.

In this sentence, it means that the speaker would want or choose to have a cup of coffee in that situation.

With food and drink, Danish have can correspond to several natural English translations depending on context:

• have
• take
• want
• would like

So Jeg ville have en kop kaffe can feel close to I’d have a cup of coffee.

If you want a more explicitly polite or desire-based phrase, Danish often uses gerne:

• Jeg ville gerne have en kop kaffe = I would like a cup of coffee

In your sentence, gerne is not necessary, because this is a hypothetical statement, not a request.

Here you need hvis because this is a condition.

• hvis = if in the sense of on the condition that
• om = if/whether in indirect questions

Examples:

• Hvis jeg har tid, kommer jeg. = If I have time, I’ll come.
• Jeg ved ikke, om jeg har tid. = I don’t know if/whether I have time.

So in your sentence, hvis is the correct choice.

Because Danish usually does not use a separate word corresponding to English of in expressions like this.

So Danish says:

• en kop kaffe = a cup of coffee
• et glas vand = a glass of water
• en flaske vin = a bottle of wine

Also, kop is a common-gender noun, so it takes en:

• en kop
• not et kop

Yes. You can also say:

Hvis jeg ikke skulle op tidligt i morgen, ville jeg have en kop kaffe.

That is perfectly natural.

But when the hvis clause comes first, the main clause still follows Danish verb-second word order, so you get:

• ville jeg have ...

not:

• jeg ville have ...

So the two patterns are:

• Jeg ville have en kop kaffe, hvis ...
• Hvis ..., ville jeg have en kop kaffe

Both are correct.