У магазині дев’ять вікон.

Because after у meaning in, Ukrainian uses the locative case for the noun.

So:

• магазин = store dictionary form
• у магазині = in the store

This is a very common pattern:

• у місті = in the city
• у будинку = in the building
• у магазині = in the store

Yes. У and в often both mean in.

The choice is usually made for ease of pronunciation and rhythm. Ukrainian often alternates у and в to avoid awkward sound combinations.

So both of these can be natural:

• У магазині дев’ять вікон
• В магазині дев’ять вікон

The version with у is completely normal here.

It can be there, but it does not have to be.

Ukrainian often omits the present-tense to be in simple descriptive sentences. So:

• У магазині дев’ять вікон = natural
• У магазині є дев’ять вікон = also natural, a bit more explicit

Also, Ukrainian does not need a dummy word like English there in there are. The sentence is structured more like:

• In the store, nine windows

with the idea of are understood.

Because after дев’ять and other numbers 5 and higher, Ukrainian usually uses the genitive plural of the noun.

So:

• вікно = window
• вікна = windows after 2, 3, 4 in many counting patterns
• вікон = genitive plural, used after 5+

Examples:

• два вікна = two windows
• чотири вікна = four windows
• дев’ять вікон = nine windows

This is one of the most important number patterns in Ukrainian.

Дев’ять is the cardinal number nine. It is the word that controls the form of вікон.

In other words:

• the number is дев’ять
• the counted noun must appear in the form required by that number
• for 5 and above, that form is usually genitive plural

So дев’ять вікон is a standard counting phrase meaning nine windows.

A rough English-friendly pronunciation is DEV-yat’.

The apostrophe in дев’ять is important. It shows that:

• the в is not softened
• the я begins with a y sound, like ya

So it is not said as one blended soft syllable. It is more like:

• dev + yat’

The final ь is the soft sign, which softens the final т.

Yes. Ukrainian word order is fairly flexible.

У магазині дев’ять вікон is very natural because it starts with the location and then gives the information about it.

You could also say:

• Дев’ять вікон у магазині

But that changes the emphasis a little. It puts more focus on дев’ять вікон.

So the original sentence feels like:

• As for the store, it has nine windows
• or In the store, there are nine windows

Because Ukrainian has no articles.

There is no direct equivalent of English a or the in normal Ukrainian grammar. Whether something is a store or the store is understood from context.

So магазин can mean:

• a store
• the store

and у магазині can mean:

• in a store
• in the store

depending on the situation.