В нашем городе есть маленький книжный магазин рядом с метро.

Russian usually does not use the verb иметь with inanimate subjects (like cities, houses, etc.) to express possession in this way.

To say There is / Our city has, Russian normally uses an existential construction:

• В нашем городе есть магазин. – In our city there is a shop / Our city has a shop.

So the structure is:

• В нашем городе – in our city (place, prepositional case)
• есть – there is/are (existence)
• маленький книжный магазин – the thing that exists (subject in nominative)

Наш город имеет маленький книжный магазин is grammatically possible, but it sounds unnatural and overly formal in everyday speech. Native speakers strongly prefer В нашем городе есть… here.

Because of the preposition в meaning in (location).

With в meaning “in/inside” (where?), Russian uses the prepositional case:

• город (nominative) → в городе (prepositional)
• дом → в доме
• школа → в школе

The typical prepositional ending for singular nouns is -е (or -и after some consonants). So:

• город
  • в (where?) → в городе.

The word наш (our) is a pronoun-adjective and must agree with городе in:

• gender: masculine
• number: singular
• case: prepositional

The forms of наш (masculine singular) are:

• Nominative: наш
• Genitive: нашего
• Dative: нашему
• Accusative: наш / нашего
• Instrumental: нашим
• Prepositional: о нашем / в нашем

Since городе is prepositional (after в meaning “in”), the correct matching form is нашем:

• в нашем городе – in our city

в наш город would be accusative (into our city, direction), not in our city. That would answer куда? (where to?), not где? (where?).

Есть is often used in Russian existential sentences, but it is not always required.

Your sentence:

• В нашем городе есть маленький книжный магазин рядом с метро.

is completely standard and neutral.

You can also say:

• В нашем городе маленький книжный магазин рядом с метро.

This version sounds a bit more like you are describing the city (almost like “our city is such that it has a small bookstore by the metro”), rather than simply stating bare existence.

General idea:

• With есть: neutral, factual there exists.
• Without есть: often a bit more descriptive, sometimes slightly more stylistic or colloquial.

Both are correct; the version with есть is the safest for learners.

The phrase маленький книжный магазин is in the nominative case, because it is the subject of the existential construction.

Structure:

• В нашем городе – where? (prepositional, setting the location)
• есть – there is
• маленький книжный магазин – what exists there? (subject, nominative)

The noun:

• магазин – masculine, singular, nominative

The adjectives must agree with магазин:

• masculine
• singular
• nominative

So we get:

• маленький (not маленькая / маленькое / маленькие)
• книжный (not книжная / книжное / книжные)

Hence: маленький книжный магазин.

In Russian, the gender of a noun is mostly determined by its ending in the nominative singular:

• Masculine: usually ends in a consonant (e.g. магазин, дом, стол)
• Feminine: usually ends in -а / -я (e.g. книга, машина, неделя)
• Neuter: usually ends in -о / -е (e.g. окно, море)

Since магазин ends in a consonant, it is masculine.

Therefore, the adjective must use the masculine form:

• Masculine: маленький магазин
• Feminine: маленькая книга
• Neuter: маленькое окно

Literally:

• книжный магазин = book shop (book + adjectival ending → “book-type shop”)
• магазин книг = shop of books

In practice:

• книжный магазин is the standard, natural way to say bookstore / bookshop.
• магазин книг is also understandable, but it sounds more formal, descriptive, or slightly unusual as the normal name of the place.

Compare in English:

• shoe store vs. store of shoes
  The first is what people normally say; the second is grammatical but less idiomatic.

So you should learn книжный магазин as the normal fixed expression for bookshop.

Russian allows quite a bit of word order flexibility. Some versions sound more natural than others.

Natural variants include:

• В нашем городе есть маленький книжный магазин рядом с метро.
• В нашем городе рядом с метро есть маленький книжный магазин.
• Рядом с метро в нашем городе есть маленький книжный магазин.

These differ mainly in emphasis (what is presented as “new” or important information).

Есть маленький книжный магазин в нашем городе is grammatically possible, but it usually sounds like you are answering a question such as Где есть маленький книжный магазин? – В нашем городе.
So it can be used, but the neutral starting point is the original word order.

The version you gave:

• В нашем городе маленький книжный магазин есть рядом с метро

is technically possible but sounds awkward and unnatural in everyday speech.

The preposition рядом с means next to / beside / right by.

In Russian, с in this meaning (with / next to) requires the instrumental case:

• рядом с домом – next to the house
• рядом с парком – next to the park
• рядом с другом – next to a friend

So grammatically, метро here is also in the instrumental case: рядом с (чем?) метро.

However, the noun метро is indeclinable: its form does not change in different cases. So:

• Nominative: метро
• Genitive: метро
• Dative: метро
• Instrumental: метро
• Prepositional: метро

The case changes, but the form stays the same.

All three can mean near the metro, but there are nuances:

• рядом с метро – right next to the metro, very close, beside it. Often implies immediate proximity.
• возле метро – near the metro, fairly close, but not necessarily directly at the entrance.
• около метро – around / in the area of the metro, sometimes a bit looser, “in the vicinity”.

In many everyday situations, they can all be translated simply as near the metro, and all are acceptable.

If you want to emphasize “right by the metro station”, рядом с метро (or прямо у метро) is a good choice.

Russian often uses метро by itself to mean the metro system / metro station area, similar to how English sometimes says at the bus stop vs at the bus.

In everyday speech, рядом с метро is naturally understood as near the metro station.

If you want to be more explicit, you can say:

• рядом со станцией метро – near the metro station
• рядом с метро станция would be wrong; станция метро is the fixed collocation.

Note also:

• Russian has no articles (no a, an, the), so we just say метро, not “the metro” or “a metro”.

Yes, that is correct Russian, but the meaning changes slightly:

• В нашем городе есть маленький книжный магазин…
  → The shop is small.
• В нашем маленьком городе есть книжный магазин…
  → The city is small.

In the second sentence:

• нашем маленьком городе – both нашем and маленьком modify городе (our small city)
• книжный магазин – just “bookstore” (no size mentioned)

So be careful which noun your adjective is attached to.

Stressed vowels (the ones you pronounce more strongly) are marked:

• В на́шем го́роде есть ма́ленький кни́жный магази́н ря́дом с метро́.

Word by word:

• В – [v]
• на́шем – НА-shem (stress on first syllable)
• го́роде – ГО-ro-de
• есть – [yest’] (single syllable)
• ма́ленький – МА-len’-kiy (soft л: tongue towards the palate)
• кни́жный – КНИ-zhnyy (the жн sounds like [zh-n])
• магази́н – ma-ga-ЗИН
• ря́дом – РЯ-dom (rolled or tapped р, soft р, like r’)
• с – [s]
• метро́ – me-ТРО

Rhythmically: В НА́шем ГО́роде есть МА́ленький КНИ́жный магази́н РЯ́дом с метро́.

In the present, we use есть:

• В нашем городе есть маленький книжный магазин рядом с метро.

In the past, we normally use был / была / было / были instead of есть:

• В нашем городе был маленький книжный магазин рядом с метро.
  – There was a small bookstore in our city near the metro.

In the future, we use будет / будут:

• В нашем городе будет маленький книжный магазин рядом с метро.
  – There will be a small bookstore in our city near the metro.

So the pattern is:

• Past: был / была / было / были
• Present: usually есть (often omitted in other types of sentences)
• Future: будет / будут