A király a történetben van.

No. In Hungarian, a (or az) is the definite article, so it means the, not a/an.

So:

• a király = the king
• a történetben = in the story

If you wanted a king, Hungarian would usually use egy:

• Egy király a történetben van. = A king is in the story.
  More naturally: Van egy király a történetben. = There is a king in the story.

Because both nouns are definite:

• a király = the king
• a történetben = in the story

Hungarian uses the article before the noun even when that noun also has a case ending attached to it. So a történetben is literally:

• a = the
• történetben = in story

Together: in the story

It is made of:

• történet = story
• -ben = in

So:

• történetben = in the story/story-in depending on how literally you analyze it

Hungarian often uses case endings where English uses prepositions. Instead of a separate word like in, Hungarian attaches the meaning to the noun.

Because of vowel harmony, which is a major feature of Hungarian.

Hungarian suffixes often come in different forms, and the form depends on the vowels in the word.

Here:

• történet has front vowels
• so the suffix is -ben

Compare:

• házban = in the house
• kertben = in the garden

So -ban/-ben both mean in, but the word chooses the form that fits its vowels.

van is the 3rd person singular form of the verb lenni, which means to be.

Here, van means is:

• A király a történetben van. = The king is in the story.

In this kind of sentence, van shows location or presence.

That is a very common and very good question.

Hungarian often omits van in the present tense when the predicate is a noun or adjective:

• A király öreg. = The king is old.
• A király hős. = The king is a hero.

But with location, Hungarian normally keeps van:

• A király a történetben van. = The king is in the story.
• A király a várban van. = The king is in the castle.

So the short version is:

• adjective/noun predicate: often no van
• location/existence: van is usually needed

Because that is a natural Hungarian word order for this kind of sentence.

Hungarian word order is more flexible than English word order, but it is not random. In a neutral sentence, the part closely connected to the verb often comes right before it. Here:

• a király = topic, what we are talking about
• a történetben van = what we say about the king

So A király a történetben van sounds natural.

If you change the order, the emphasis can change. Hungarian word order often shows focus or what is being highlighted, not just grammar.

Yes, but the nuance may change.

For example:

• A király a történetben van.
  Neutral: The king is in the story.

• A történetben van a király.
  This can put more emphasis on in the story or sound more contrastive, depending on context.

Also, if you want to say There is a king in the story, Hungarian more naturally says:

• Van egy király a történetben.

So different word orders are possible, but they are not always interchangeable in tone or emphasis.

Because the subject form in Hungarian is usually just the basic dictionary form, with no extra ending. This is often called the nominative, and in Hungarian it is typically a zero ending.

So:

• király = king
• in this sentence, it is also the subject: the king

Hungarian often marks other roles with suffixes, but the plain subject form usually stays unchanged.

Both a and az mean the. The choice depends on the next word:

• a before a consonant sound
• az before a vowel sound

So here we get:

• a király because király starts with a consonant
• a történetben because történetben also starts with a consonant

Compare:

• az alma = the apple
• az ember = the person

So a and az are just two forms of the same article.