Jeg vil gerne have sådan et gavekort til min søster.

Gerne makes the sentence sound more polite and natural.

• Jeg vil have ... can sound quite direct: I want ...
• Jeg vil gerne have ... is softer: I would like ...

So in shops, cafés, or service situations, vil gerne is very common.

In this sentence, gerne does not mean gladly in the literal English sense. It helps express a polite wish.

So:

• jeg vil gerne have = I would like
• literally, it is something like I want gladly to have, but that is not how you should translate it in normal English

It is best to learn vil gerne as a very common polite pattern.

Not always. In Danish, have often appears in expressions where English would use have, get, buy, or take depending on context.

Here, jeg vil gerne have sådan et gavekort means something like:

• I’d like a gift card like that
• I’d like to get a gift card like that

So have is part of the standard expression, even if English would not always use have in the same situation.

Sådan et means such a or one like that / one of that kind.

In this sentence, it points to a type of gift card:

• sådan et gavekort = that kind of gift card / a gift card like that

It often refers to something already visible or already mentioned.

Because gavekort is a singular noun, and Danish normally needs an article with it here.

• et gavekort = a gift card
• sådan et gavekort = such a gift card / a gift card like that

The et matches the noun gavekort, which is a neuter noun.

Because gavekort is a neuter noun in Danish.

Danish indefinite singular articles are:

• en for common gender nouns
• et for neuter nouns

So:

• et gavekort
• gavekortet = the gift card

Unfortunately, grammatical gender usually has to be learned with each noun.

Yes. Then sådan et would become sådan en.

For example:

• sådan en bog = such a book
• sådan et gavekort = such a gift card

So the form changes to match the gender of the noun.

Because Danish, like English, often forms compound nouns, but Danish does this even more consistently.

• gave = gift
• kort = card
• gavekort = gift card

In Danish, these compounds are usually written as one word.

Here, til means for / to, in the sense of intended for someone.

• et gavekort til min søster = a gift card for my sister

In many cases where English uses for, Danish uses til when something is meant for a person.

For does exist in Danish too, but it is not the natural choice in this sentence.

Because in Danish, possessives like min, din, hans, hendes normally replace the definite ending.

So you say:

• min søster = my sister
• not min søsteren

Compare:

• søster = sister
• søsteren = the sister
• min søster = my sister

This is different from English only in form, not in meaning.

Yes. This is standard Danish main-clause word order:

• Jeg = subject
• vil = finite verb
• gerne have = rest of the verb expression
• sådan et gavekort = object
• til min søster = prepositional phrase

So the pattern is very natural: Subject + verb + adverb + infinitive/object/etc.

In Danish main questions, the verb usually comes before the subject.

So:

• Jeg vil gerne have sådan et gavekort til min søster.
• Vil du gerne have sådan et gavekort til din søster?

If you want the exact same person:

• Vil jeg gerne have sådan et gavekort til min søster?
  This is grammatical, but it sounds unusual unless you are asking something very specific.

A more natural shop question would be:

• Vil du gerne have sådan et gavekort til din søster?
  = Would you like a gift card like that for your sister?

Yes, it is polite and natural.

Jeg vil gerne have ... is a very common way to ask for something in Danish. It is not overly formal, but it is courteous.

You could also hear:

• Jeg vil gerne købe ... = I would like to buy ...
• Kan jeg få ...? = Can I have / get ...?

All of these are natural, but jeg vil gerne have is very common and useful to learn.