Ez az új sisak biztonságos.

Because Hungarian normally uses both a demonstrative and the definite article before the noun.

So:

• ez = this
• az = the here

In standard Hungarian, this helmet is:

• ez a sisak or ez az ... before a vowel sound

In your sentence:

• Ez az új sisak = This new helmet

So the pattern is very common:

• ez a ház = this house
• ez az autó = this car

The article is not optional in normal standard usage when a noun follows the demonstrative.

Hungarian has two forms of the definite article:

• a
• az

Which one you use depends on the following sound:

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

Since új begins with a vowel sound, Hungarian uses:

• az új sisak

Compare:

• a sisak = the helmet
• az új sisak = the new helmet

This is similar to the old English distinction between a and an.

In Hungarian, the verb to be is usually left out in the present tense when the sentence is:

• third person, and
• describing something with a noun or adjective

So English:

• This new helmet is safe.

Hungarian:

• Ez az új sisak biztonságos.

No van is needed here.

But in other contexts, van does appear. For example:

• A sisak a táskában van. = The helmet is in the bag.

So in your sentence, omitting is is completely normal.

Because they are doing two different jobs.

• új is an adjective inside the noun phrase: the new helmet
• biztonságos is the predicate adjective, telling you something about the helmet: is safe

So:

• az új sisak = the new helmet
• biztonságos = safe

Put together:

• Ez az új sisak biztonságos. = This new helmet is safe.

English does the same thing:

• the new helmet → adjective before the noun
• the helmet is safe → adjective after is

Hungarian just leaves out is here.

Yes. That would mean:

• The new helmet is safe.

The difference is:

• Ez az új sisak biztonságos. = This new helmet is safe.
• Az új sisak biztonságos. = The new helmet is safe.

So ez adds the idea of this.

Sisak is in the nominative singular, which is the basic dictionary form.

Hungarian often adds endings for cases, possession, plurality, and so on, but here none are needed because sisak is simply the subject of the sentence:

• Ez az új sisak = This new helmet

So the bare form sisak is exactly what you would expect.

Hungarian word order is more flexible than English, but different orders change the focus or emphasis.

The neutral sentence is:

• Ez az új sisak biztonságos.

That is a natural, straightforward way to say This new helmet is safe.

You may also hear other orders in special contexts, but they usually sound like they are stressing something:

• Biztonságos ez az új sisak.
  = This new helmet is safe / Safe is what this new helmet is.

So yes, word order can change, but the version you were given is the most neutral and learner-friendly.

A rough pronunciation guide for English speakers is:

• Ez ≈ ez
• az ≈ awz / uz depending on accent, but keep it short
• új ≈ oo-y
• sisak ≈ shee-shok
• biztonságos ≈ beez-ton-shaa-gosh

A few useful points:

• Hungarian s sounds like English sh
• Hungarian sz sounds like English s
• ú is a long vowel, so új starts with a long oo
• Stress is always on the first syllable of each word:
  • EZ
  • AZ
  • Új
  • SI-sak
  • BIZ-ton-sá-gos

Usually yes, in this kind of sentence.

• biztonságos = safe, secure

For an object like a helmet, it means it is safe or provides safety.

The word comes from:

• biztonság = safety, security
• -os / -s = an adjective-forming ending in this case

So biztonságos literally has the idea of safety-related / safety-providing / secure.

They have different structures and meanings:

• Ez egy új sisak. = This is a new helmet.
• Ez az új sisak biztonságos. = This new helmet is safe.

In the first sentence:

• ez stands alone as this
• egy új sisak means a new helmet

In the second sentence:

• ez az új sisak is one noun phrase: this new helmet
• biztonságos tells you something about it

So one sentence identifies the object, and the other describes it.

In standard Hungarian, yes.

With a noun, the normal pattern is:

• ez a ... / ez az ... = this ...
• az a ... / az az ... = that ...

Examples:

• ez a könyv = this book
• ez az alma = this apple
• az a ház = that house

So in your sentence, ez az új sisak is exactly the expected pattern.

For English speakers, this feels unusual at first, but it is a very basic and common feature of Hungarian grammar.