Na stole brakuje jednego sztućca.

In this sentence, brakuje is being used in a very common impersonal pattern meaning there is a lack of / ... is missing.

So Na stole brakuje jednego sztućca is structured more like:

On the table, there is a lack of one utensil.

Because of that, the verb stays in the fixed 3rd person singular form brakuje. It does not behave like a normal verb with a clear nominative subject.

This is why Polish often says:

• Brakuje cukru. = There is no sugar / Sugar is missing.
• Brakuje jednego krzesła. = One chair is missing.

Because the verb brakować / brakuje commonly takes the thing missing in the genitive.

So:

• jeden sztuciec = nominative
• jednego sztućca = genitive

That is why the sentence is not brakuje jeden sztuciec, but brakuje jednego sztućca.

This is a very important pattern to remember:

• Brakuje mleka.
• Brakuje czasu.
• Brakuje jednej osoby.

All of the missing things are in the genitive.

Na stole is in the locative case.

The preposition na can take different cases depending on meaning:

• na + accusative for movement onto something
  
  • Kładę książkę na stół. = I put the book onto the table.
• na + locative for location on something
  
  • Książka leży na stole. = The book is lying on the table.

Here the meaning is location, so we use na stole = on the table.

The dictionary form is sztuciec.

Its forms here are:

• sztuciec = nominative singular
• sztućca = genitive singular

So jednego sztućca literally means of one utensil / one piece of cutlery.

Yes. This is a point that often confuses learners.

• sztućce usually means cutlery as a set or category
• sztuciec means one item of cutlery, such as a fork, knife, or spoon

So in this sentence, jednego sztućca means that one individual item is missing.

If you wanted to be more specific, you could also say:

• Brakuje jednego widelca. = One fork is missing.
• Brakuje jednego noża. = One knife is missing.
• Brakuje jednej łyżki. = One spoon is missing.

Because jednego has to match the case of sztućca, which is genitive.

The adjective/numeral jeden changes for case, gender, and number, just like other Polish adjectives.

Here:

• jeden sztuciec = nominative
• jednego sztućca = genitive

So both words change together.

Yes, but it is not exactly the same in feel.

• Na stole brakuje jednego sztućca sounds like one piece of cutlery is missing from the table / the place setting is incomplete
• Na stole nie ma jednego sztućca simply means one piece of cutlery is not on the table

So brakuje often suggests an expected item is absent. It highlights a lack.
Nie ma is more neutral and just states absence.

Yes. Polish word order is flexible, although the neutral order here is very natural:

Na stole brakuje jednego sztućca.

Other orders are possible, for example:

• Jednego sztućca brakuje na stole.
• Brakuje jednego sztućca na stole.

But these usually sound more marked or put emphasis somewhere different. For a learner, the original version is the safest and most natural in ordinary speech.

In this sentence, na stole means on the table literally.

Polish na often corresponds to English on, and stole is the locative of stół = table.

If you wanted at the table in the sense of people sitting there, Polish often uses przy stole instead:

• Siedzimy przy stole. = We are sitting at the table.

So:

• na stole = on the table
• przy stole = at the table

Yes, very common.

The pattern is:

[place / context] + brakuje + genitive

Examples:

• W lodówce brakuje mleka. = There is no milk in the fridge / Milk is missing from the fridge.
• W klasie brakuje jednej osoby. = One person is missing from the classroom.
• Na półce brakuje jednej książki. = One book is missing from the shelf.

So this sentence is a very useful model to learn.