Zużyta bateria jest odpadem niebezpiecznym.

Here zużyta means used up / spent / no longer usable.

With batteries, English can use several words depending on context:

• used battery
• spent battery
• dead battery

In an environmental or recycling context, zużyta bateria usually means a battery that has finished its useful life and should be disposed of properly.

Because it has to agree with bateria.

In Polish, adjectives change their form to match the noun in:

• gender
• number
• case

Bateria is:

• feminine
• singular
• nominative here

So the adjective also takes the feminine singular nominative form:

• zużyty — masculine
• zużyta — feminine
• zużyte — neuter

For example:

• zużyty akumulator
• zużyta bateria
• zużyte urządzenie

Because Polish has no articles.

So bateria can mean:

• a battery
• the battery

The context tells you which one is meant. In this sentence, it is most naturally understood as a general statement: a used battery is hazardous waste.

Because after jest with a predicate noun, Polish normally uses the instrumental case.

So:

• odpad → odpadem
• niebezpieczny → niebezpiecznym

That is why the sentence says:

• jest odpadem niebezpiecznym

not

• jest odpad niebezpieczny

A very useful contrast:

• Bateria jest zużyta. — adjective only, so zużyta stays nominative
• Bateria jest odpadem niebezpiecznym. — noun phrase after jest, so it goes into the instrumental

You are right that adjectives usually come before the noun in Polish:

• niebezpieczny odpad
• niebezpiecznym odpadem

But in official, technical, or legal-style language, the adjective often comes after the noun:

• odpad niebezpieczny
• odpadem niebezpiecznym

This sounds more like a classification label, similar to saying waste classified as hazardous.

So in this sentence, the order is natural for formal or informational wording. A more neutral everyday order would be:

• Zużyta bateria jest niebezpiecznym odpadem.

Both are grammatical, but the given version sounds especially like signage, regulation language, or an informational notice.

Normally, no.

In a full standard sentence, Polish usually keeps jest:

• Zużyta bateria jest odpadem niebezpiecznym.

Leaving it out would sound incomplete, headline-like, or non-standard in normal prose.

So unlike some other Slavic languages, Polish normally does use the present-tense form of to be in sentences like this.

Yes. Polish word order is fairly flexible, though different orders change the emphasis.

The given sentence:

• Zużyta bateria jest odpadem niebezpiecznym.

is a neutral, natural statement.

Other possible versions:

• Odpadem niebezpiecznym jest zużyta bateria.
  Emphasizes the classification hazardous waste.

• Zużyta bateria to odpad niebezpieczny.
  Very common in explanations and definitions; slightly different structure, but same basic meaning.

So the original order is not the only possible one, but it is perfectly natural.

Not always. Bateria has several meanings in Polish, including:

• battery
• artillery battery
• sometimes tap/faucet fixture in everyday Polish

But in this sentence, because of zużyta and the waste/recycling context, it clearly means battery.

Also, note this useful distinction:

• bateria — an ordinary battery
• akumulator — a rechargeable battery, especially a car battery

So for a dead AA battery, bateria is exactly what you would expect.

A rough pronunciation guide:

• Zużyta bateria jest odpadem niebezpiecznym.
• zoo-ZHI-ta ba-TYE-r'ya yest ot-PA-dem nye-bes-PYECH-nim

Very roughly by word:

• zużyta — zoo-ZHI-ta
• bateria — ba-TYE-r'ya
• jest — yest
• odpadem — ot-PA-dem
• niebezpiecznym — nye-bes-PYECH-nim

A few sound notes:

• ż sounds like the s in measure
• y has no exact English equivalent; it is not the same as ee
• cz sounds like ch in church
• nie at the start of niebezpiecznym sounds like nye

You would say:

• Zużyte baterie są odpadami niebezpiecznymi.

Changes to notice:

• zużyta → zużyte
• bateria → baterie
• jest → są
• odpadem → odpadami
• niebezpiecznym → niebezpiecznymi

This is a good example of how many parts of a Polish sentence change to match number and case.