Η σφολιάτα αυτή έχει γέμιση με τυρί και είναι πιο νόστιμη από το κουλουράκι.

In Greek, the demonstrative αυτή can go either before or after the noun.

Both of these are correct:

• αυτή η σφολιάτα
• η σφολιάτα αυτή

The first one is usually the more neutral and common everyday order. The second one, used in your sentence, can sound a bit more pointed, like this particular puff pastry.

Σφολιάτα literally refers to puff pastry, but in real usage it can also mean a puff-pastry snack/pie made from that dough.

In this sentence, because it has cheese filling, it clearly means the finished food item, not just the dough.

Because Greek nouns have grammatical gender.

• η σφολιάτα is feminine
• το κουλουράκι is neuter

The article has to match the noun’s gender. That also affects other words connected to the noun:

• η σφολιάτα αυτή
• πιο νόστιμη

All of those are feminine singular because they refer to σφολιάτα.

Αυτή is the feminine singular nominative form of the demonstrative this.

It agrees with σφολιάτα, which is also feminine singular.
Compare:

• αυτός = masculine
• αυτή = feminine
• αυτό = neuter

So:

• αυτός ο καφές = this coffee
• αυτή η σφολιάτα = this puff pastry
• αυτό το κουλουράκι = this cookie/biscuit

Literally, it means has filling with cheese.

Yes, that is natural Greek, especially when describing food. It presents the filling as a feature of the item.

• γέμιση = filling
• με τυρί = with cheese

A slightly different way to say something similar would be:

• είναι γεμισμένη με τυρί = it is filled with cheese

Your sentence focuses more on the noun γέμιση than on the passive idea filled.

Because in Greek, ingredient or material nouns often appear without an article when speaking generally.

So:

• με τυρί = with cheese
• με σοκολάτα = with chocolate
• με ζάχαρη = with sugar

If you meant a specific cheese, you might use an article or a longer phrase, but here it just means cheese in general.

This is the normal Greek way to make a comparison:

πιο + adjective + από = more + adjective + than

So:

• πιο νόστιμη = tastier / more tasty
• από το κουλουράκι = than the cookie/biscuit

Together:

• πιο νόστιμη από το κουλουράκι = tastier than the cookie/biscuit

Greek also has a more compact comparative form in some cases, so you could also see:

• νοστιμότερη από ...

But πιο νόστιμη is very common and natural.

Because the adjective must agree with the noun it describes.

The thing being described is η σφολιάτα, which is feminine singular, so the adjective is also feminine singular:

• νόστιμος = masculine
• νόστιμη = feminine
• νόστιμο = neuter

So here:

• η σφολιάτα είναι νόστιμη

Because the subject is understood from the first clause.

The sentence is:

• Η σφολιάτα αυτή έχει γέμιση με τυρί
• και είναι πιο νόστιμη από το κουλουράκι

The subject of είναι is still η σφολιάτα αυτή. Greek, like English, often avoids repeating the subject when it is already clear.

Here από means than in a comparison.

So:

• πιο νόστιμη από το κουλουράκι = tastier than the cookie/biscuit

More generally, από usually means from, but in comparisons it very often means than.

Also, in Modern Greek, από normally takes the accusative case. With a neuter noun like κουλουράκι, the nominative and accusative look the same, so you see το κουλουράκι.

Κουλουράκι usually means a small biscuit/cookie, often a traditional Greek one.

The ending -άκι is a very common diminutive ending in Greek, often giving the idea of something small or familiar.

So even if a translation gives cookie or biscuit, the Greek word has a specifically Greek feel and often refers to a small baked treat.