В булочной были слойки и сладкий рулет.

Because в means location here, and Russian uses the prepositional case after в when it means in/at a place.

So:

• булочная = bakery
• в булочной = in/at the bakery

A useful contrast:

• в булочной = in/at the bakery
• в булочную = into/to the bakery

Yes. Булочная is a noun in meaning, even though it looks like an adjective in form.

This is very common in Russian: an adjective can become a noun when the main noun is understood. Historically, булочная meant something like bread shop, with the noun omitted. Over time it became a normal word for bakery or bread shop.

That is why it declines like an adjective:

• булочная
• булочной
• булочную, etc.

Russian often omits is/are in the present tense in simple sentences, but not in the past tense.

So:

• present: В булочной есть слойки... or sometimes other present-tense structures
• past: В булочной были слойки...

Here the sentence is in the past, so были is needed.

Because the sentence has two things: слойки and сладкий рулет. Together they make a compound subject, so Russian uses the plural verb.

So:

• слойки и сладкий рулет были...

Even though рулет is singular, the whole combination means pastries and a sweet roll, which is plural overall.

They are in the nominative case.

More specifically:

• слойки = nominative plural
• сладкий рулет = nominative singular

They are the things that were in the bakery, so they function as the grammatical subject/content of the existential sentence.

A helpful comparison:

• affirmative: В булочной были слойки.
• negative: В булочной не было слоек.

In the negative version, Russian often switches to the genitive.

Because сладкий agrees with рулет.

Рулет is:

• masculine
• singular
• nominative

So the adjective must also be:

• masculine
• singular
• nominative

That gives сладкий рулет.

Слойки is the plural form of слойка.

Here it is specifically nominative plural:

• singular: слойка
• plural: слойки

The ending is -и because after к, Russian writes и, not ы.

This word order is very natural in Russian for introducing what existed or was available somewhere.

The structure is basically:

• location first
• then what was there

So В булочной были слойки и сладкий рулет feels like:

• In the bakery, there were pastries and a sweet roll
• or more naturally in English, There were pastries and a sweet roll in the bakery

If you change the order to Слойки и сладкий рулет были в булочной, it is still grammatical, but the emphasis shifts more toward the pastries themselves.

Because Russian has no articles.

Russian does not have direct equivalents of English a/an and the. Whether something is definite or indefinite usually comes from context.

So this sentence could correspond to English as:

• in the bakery
• in a bakery
• the sweet roll
• a sweet roll

The exact choice depends on context, not on a separate Russian word.

Literally, в means in, so в булочной is literally in the bakery.

But in English, with shops and businesses, at is often more natural:

• at the bakery

So both can be reasonable depending on context:

• in the bakery if you are stressing physical location inside the shop
• at the bakery if you mean the place more generally