Az ár alacsony.

Hungarian has two forms of the definite article:

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

Because ár begins with the vowel á, the correct form is az ár.

In Hungarian, the verb to be is usually omitted in the present tense, third person, when the sentence links a subject to a noun or adjective.

So:

• Az ár alacsony. = The price is low.

But in other tenses, the verb appears:

• Az ár alacsony volt. = The price was low.

Both patterns exist, but they do different jobs.

• alacsony ár = low price
  Here, alacsony directly modifies the noun.

• Az ár alacsony. = The price is low.
  Here, alacsony is the predicate adjective, so it comes after the noun phrase.

So this sentence is not a noun phrase like low price; it is a full sentence.

Yes. That is grammatical, but the emphasis changes.

• Az ár alacsony. = neutral statement
• Alacsony az ár. = puts more focus on low

Hungarian word order is flexible, but it often reflects emphasis and information structure rather than just grammar.

There is no grammatical gender in Hungarian, so you do not need different forms for he/she/it or masculine/feminine.

With a singular subject, the adjective stays as alacsony:

• Az ár alacsony.

With a plural subject, the predicate adjective becomes plural:

• Az árak alacsonyak. = The prices are low.

They are related, but not the same.

• alacsony = low
• olcsó = cheap

So:

• Az ár alacsony. = The price is low.
• Ez a termék olcsó. = This product is cheap.

Hungarian often uses alacsony for levels, amounts, prices, temperatures, and similar things. Olcsó usually describes the item or service itself.

A rough guide:

• az: short vowel, roughly like oz
• ár: aahr, with a long á
• alacsony: roughly AW-law-chony

A few useful sound notes:

• Stress is always on the first syllable in Hungarian.
• á is a long vowel.
• cs sounds like English ch.
• ny sounds somewhat like the ny in canyon.

Yes, very much. In Hungarian, accents are part of the spelling and pronunciation.

• a and á are different vowels
• á is longer and also different in quality

So ár must be written with á, not plain a.

Add nem before the adjective:

• Az ár nem alacsony. = The price is not low.

That is the normal way to negate this kind of sentence.