Van egy kutya és egy macska; az utóbbi fiatal.

Because van is being used in an existential sense: there is / there are.

So:

• Van egy kutya. = There is a dog.
• Van egy kutya és egy macska. = There is a dog and a cat.

This is different from sentences like A kutya fiatal (The dog is young), where Hungarian usually does not say the present-tense is in the 3rd person.

In this kind of existential sentence, Hungarian very often uses singular van when introducing a whole situation or list as new information, even if more than one item follows.

So Van egy kutya és egy macska is natural Hungarian.

A simple way to think about it is:

• van here introduces the existence of the whole setup
• the sentence is not built exactly the same way as English there are

You will still see vannak with clearly plural subjects, for example:

• Vannak kutyák a kertben. = There are dogs in the garden.

But in this sentence, van is the normal choice.

Because egy is the Hungarian indefinite article as well as the number one.

Here it means a/an:

• egy kutya = a dog
• egy macska = a cat

Since both nouns are singular, countable, and newly introduced, each one gets its own egy.

Az utóbbi means the latter, meaning the second of the two things just mentioned.

In this sentence, the two animals are:

1. egy kutya = a dog
2. egy macska = a cat

So az utóbbi refers to the cat.

A useful pair to learn is:

• az előbbi = the former
• az utóbbi = the latter

Because utóbbi is adjective-like, and when it is used by itself to mean the latter, it normally takes the definite article.

So:

• az utóbbi = the latter

Also, the article is az rather than a because the next word begins with a vowel:

• a kutya
• az utóbbi

This is similar to English a / an.

Because in Hungarian, the verb to be is usually left out in the present tense, 3rd person when the predicate is a noun or adjective.

So:

• A kutya fiatal. = The dog is young.
• Az utóbbi fiatal. = The latter is young.

This is a very important Hungarian pattern.

But the verb does appear in other persons or tenses:

• Fiatal vagyok. = I am young.
• Fiatal volt. = He/She/It was young.

Yes, absolutely.

• A macska fiatal. = The cat is young.

That is simpler and more everyday.

Using az utóbbi is useful when you want to refer back to the second-mentioned item without repeating the noun. It can sound a bit more formal or written than just repeating a macska.

Hungarian word order is flexible, but it depends strongly on information structure.

Van egy kutya és egy macska is the most natural way to introduce these animals as new information. It works like an existential sentence: there is a dog and a cat.

Egy kutya és egy macska van is possible in some contexts, but it sounds more marked, as if you are emphasizing or contrasting that phrase.

So for a neutral sentence, Van egy kutya és egy macska is the best choice here.