У магазині дев’ятнадцять товарів.

Because у/в means in, and when it shows a location, the noun usually goes into the locative case.

• Dictionary form: магазин = store/shop
• Locative singular: магазині

So:

• у магазині = in the store
• у магазин would suggest motion into the store, not location

A useful contrast:

• Я в магазині. = I am in the store.
• Я йду в магазин. = I am going to the store.

In your sentence, the meaning is location, so магазині is correct.

Because after 19, Ukrainian normally uses the noun in the genitive plural.

The basic pattern is:

• 1
  • singular
• 2, 3, 4
  • nominative plural–type form
• 5 and higher
  • genitive plural

So:

• один товар
• два товари
• чотири товари
• п’ять товарів
• дев’ятнадцять товарів

Here:

• товар = product/item/goods
• товарів = genitive plural of товар

So дев’ятнадцять товарів is exactly the expected pattern.

Because in Ukrainian, the present-tense verb to be is often omitted.

So a sentence like:

• У магазині дев’ятнадцять товарів.

naturally means:

• There are nineteen products in the store.

You can also sometimes hear:

• У магазині є дев’ятнадцять товарів.

The word є adds an explicit idea of there is/there are or exists. But leaving it out is very normal and natural in Ukrainian.

Yes. У and в both mean in here.

Ukrainian often switches between у and в for euphony—basically, whichever sounds smoother next to the surrounding sounds.

So both of these are correct:

• У магазині дев’ятнадцять товарів.
• В магазині дев’ятнадцять товарів.

They mean the same thing. The choice is usually about pronunciation flow, not grammar or meaning.

Not completely, but the given order is very natural.

У магазині дев’ятнадцять товарів is a neutral way to present the information:

• first the location
• then the quantity and noun

You can change the order, but it changes the emphasis. For example:

• Дев’ятнадцять товарів у магазині.
  This puts more focus on nineteen products.

The original sentence sounds like a normal there are ... in ... statement, so it is a very good default pattern to learn.

Товар usually means product, item for sale, or merchandise.

In this sentence, товарів is best understood as something like:

• products
• items
• goods

One thing that may feel different from English: Ukrainian can use товар as a countable noun more easily than English uses goods. So дев’ятнадцять товарів is perfectly normal.

A rough pronunciation is:

• de-vyat-NAD-tsyat’

The apostrophe in дев’ятнадцять is important. It shows that the я is pronounced as ya, and that the previous consonant is not softened in the usual way.

So в’я is pronounced more like:

• v + ya

not like a single softened consonant.

A few useful points:

• дев’ятнадцять = nineteen
• the stress is on на: дев’ятнАдцять
• the ending -ть is soft

The apostrophe is a normal spelling feature in Ukrainian, not a quotation mark.

Because Ukrainian does not have articles like English a/an/the.

So:

• магазин can mean a store or the store
• товари / товарів can mean products, the products, etc.

Context tells you which English article makes sense.

That means У магазині дев’ятнадцять товарів does not literally mark whether English should say:

• in a store
• in the store

The surrounding context would decide that.

Not in the way adjectives do.

With numerals like дев’ятнадцять, the normal pattern is:

• the numeral stays in its numeral form
• the noun after 5+ goes into genitive plural

So you do not try to make дев’ятнадцять agree with товарів the way you would with an adjective.

Also, дев’ятнадцять does not change for gender here. You use the same numeral with masculine, feminine, or neuter nouns in this kind of construction:

• дев’ятнадцять товарів
• дев’ятнадцять книг
• дев’ятнадцять вікон

The noun changes form; the numeral pattern stays the same.