Annak ellenére, hogy az elem kicsi, veszélyes hulladék.

Yes. Annak ellenére, hogy is a common expression meaning although, even though, or despite the fact that.

In this sentence, it introduces a contrast:

• az elem kicsi = the battery is small
• veszélyes hulladék = it is hazardous waste

So the whole pattern is:

Annak ellenére, hogy + clause, main clause

A natural English equivalent is Although the battery is small, it is hazardous waste.

Because ellenére requires the dative form.

The basic pattern is:

• valaminek ellenére = despite something

So:

• az = that
• annak = of that / to that / that in the -nak/-nek form

That is why Hungarian says annak ellenére.

You can think of the structure as roughly:

• despite that...

and then the hogy clause explains what that that refers to.

Ellenére is a postposition. Hungarian often uses postpositions where English uses prepositions.

So English says:

• despite that

but Hungarian literally uses something more like:

• that-despite

That is why it comes after annak, not before it.

This is normal in Hungarian:

• a ház mellett = beside the house
• a ház előtt = in front of the house
• annak ellenére = despite that

Hungarian has two forms of the definite article:

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

Since elem begins with the vowel e, the correct form is:

• az elem

Compare:

• a ház = the house
• az autó = the car
• az elem = the battery

It can mean both, depending on context.

Common meanings of elem include:

• element
• battery / cell

In this sentence, because it is described as veszélyes hulladék and is something small that needs disposal, battery is the natural meaning.

In everyday Hungarian, elem often means a small battery, especially a disposable one. A rechargeable battery is more often akkumulátor or informally akku.

Yes—this is an important difference.

In the sentence:

• az elem kicsi

kicsi is not directly modifying the noun inside one noun phrase. It is part of the predicate, so it means:

• the battery is small

When the adjective directly describes the noun, it normally goes before the noun:

• kis elem / kicsi elem = small battery

So:

• az elem kicsi = the battery is small
• kis elem = small battery

Hungarian often leaves out the verb to be in the present tense, third person when the predicate is a noun or adjective.

So:

• az elem kicsi literally looks like the battery small
• (az elem) veszélyes hulladék literally looks like (the battery) hazardous waste

But both mean is in English.

This is called the zero copula.

Compare:

• Az elem kicsi. = The battery is small.
• Az elem veszélyes hulladék. = The battery is hazardous waste.

But in other persons or tenses, the verb appears:

• Én kicsi vagyok. = I am small.
• Az elem kicsi volt. = The battery was small.

Because Hungarian often leaves out a repeated subject when it is already clear from context.

The sentence really means:

• Annak ellenére, hogy az elem kicsi, (az elem) veszélyes hulladék.

The second az elem is simply omitted because it would be repetitive.

This is very natural in Hungarian. English usually needs it, but Hungarian often does not.

Also, Hungarian does not use gendered subject pronouns like he/she/it in the same way English does.

Because here veszélyes hulladék is a predicate noun naming what kind of thing the battery is.

Hungarian often omits the article in this kind of classification:

• Péter orvos. = Peter is a doctor.
• Ez probléma. = This is a problem.
• Az elem veszélyes hulladék. = The battery is hazardous waste.

Using egy here would sound less natural, because the point is not one hazardous waste item, but rather the category: it counts as hazardous waste.

Because Hungarian normally sets off subordinate clauses with commas.

Here, hogy az elem kicsi is part of the concessive structure introduced by annak ellenére.

So the standard punctuation is:

• Annak ellenére, hogy az elem kicsi, veszélyes hulladék.

You get:

1. a comma before hogy
2. a comma after the subordinate clause, before the main clause

This is standard Hungarian punctuation.

Yes to both.

A shorter alternative is:

• Bár az elem kicsi, veszélyes hulladék.

Here bár means although. This version is simpler and very common.

The original annak ellenére, hogy is a bit longer and can sound a little more formal or emphatic, like despite the fact that.

Word order can also change, for example:

• Az elem annak ellenére, hogy kicsi, veszélyes hulladék.

This is also correct. It puts az elem first and inserts the concessive part in the middle.

Hungarian word order is more flexible than English, and different orders can shift emphasis slightly, but the basic meaning stays the same.