A magánútlevél érvényes, és az érvényességi idő hosszú.

Hungarian usually omits the present-tense verb van (to be) in the 3rd person singular and plural when the predicate is a noun or adjective.

So:

• A magánútlevél érvényes. = The private passport is valid.
• Az érvényességi idő hosszú. = The validity period is long.

This is completely normal Hungarian.
But if you change the tense, the verb appears:

• A magánútlevél érvényes volt. = The private passport was valid.

And with I/you, the verb also appears:

• Érvényes vagyok would be I am valid.

Because here érvényes is not directly describing the noun inside the noun phrase; it is the predicate adjective.

Compare:

• az érvényes magánútlevél = the valid private passport
  
  • adjective before the noun
• A magánútlevél érvényes. = The private passport is valid.
  
  • adjective after the noun, because it is part of the predicate

The same pattern appears in the second clause:

• a hosszú érvényességi idő = the long validity period
• Az érvényességi idő hosszú. = The validity period is long.

Both a and az mean the. The choice depends on the following sound:

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

So:

• a magánútlevél because magánútlevél begins with m
• az érvényességi idő because érvényességi begins with é, a vowel

This is similar to English a/an, but Hungarian does it with the definite article.

They are related, but they do different jobs.

• érvényes = valid
  This is a normal adjective.
• érvényességi = roughly relating to validity or validity-
  This is a derived form used before another noun.

So:

• A magánútlevél érvényes. = The passport is valid.
• érvényességi idő = validity period

You cannot usually swap them:

• érvényes idő would not be the normal way to say validity period
• érvényességi útlevél would also be wrong here

It is a very typical Hungarian compound-like structure.

Breakdown:

• érvény = validity / force / effect
• érvényes = valid
• érvényesség = validity
• érvényességi = of validity / validity-related
• idő = time / period

So érvényességi idő literally means something like validity time/period.

A very common pattern in Hungarian is:

• noun
• derived noun
• derived adjective in -i
• then another noun

This kind of structure is extremely common in formal or administrative vocabulary.

Yes, it is one word because Hungarian very often forms compound nouns as single words.

Here:

• magán = private
• útlevél = passport

Together:

• magánútlevél = private passport

This is normal Hungarian spelling. English often writes these as two words, but Hungarian much more readily joins them into one.

Because és is joining two clauses, not just two simple words.

The clauses are:

1. A magánútlevél érvényes
2. az érvényességi idő hosszú

Each clause has its own subject and predicate, so the comma is standard:

• A magánútlevél érvényes, és az érvényességi idő hosszú.

If és just linked two words, there would usually be no comma:

• hosszú és fontos időszak = a long and important period

No, Hungarian word order is flexible, but different orders change the focus or emphasis.

The given sentence is fairly neutral:

• A magánútlevél érvényes, és az érvényességi idő hosszú.

You could also say:

• Érvényes a magánútlevél
  This puts more focus on érvényes.
• Hosszú az érvényességi idő
  This emphasizes hosszú more strongly.

So the original sentence is a natural neutral statement, while other orders may sound more contrastive or emphatic.

Yes, idő very often means time, but it can also mean a period or duration, depending on context.

In érvényességi idő, it means something like:

• period
• term
• duration

So although the literal pieces are close to validity time, the natural meaning is validity period.

This is common in Hungarian: one noun may cover several related meanings that English splits into different words.

Then the verb to be must appear, because Hungarian only omits it in the present tense in this kind of sentence.

Past tense:

• A magánútlevél érvényes volt, és az érvényességi idő hosszú volt.

Here volt means was.

So:

• present: A magánútlevél érvényes.
• past: A magánútlevél érvényes volt.

This is a useful contrast to remember.

The accent marks matter in Hungarian: they usually show vowel quality and often length too.

A few important ones here:

• á in magánútlevél: a long open a
• ú in útlevél: a long oo
• é in érvényes, érvényességi: a long ay-like sound
• í in érvényességi: a long ee
• ő in idő: a long front rounded vowel, which English does not really have

Also, every written vowel should be pronounced clearly. Hungarian spelling is much more regular than English spelling.

A rough pronunciation guide would be:

• magánútlevél ≈ ma-GAAN-oot-le-vayl
• érvényes ≈ AIR-vay-nyesh
• idő ≈ ee-doe with a rounded final vowel

The rough guide is only approximate, but it helps you notice that the accents are important and should not be ignored.