Одна кроссовка лежит под диваном.

Because кроссовка is a feminine singular noun.

In Russian, one changes for gender:

• один = masculine
• одна = feminine
• одно = neuter

So with кроссовка, you say одна кроссовка.

Yes. Кроссовка is the singular, and кроссовки is the plural.

That said, Russians often use the plural more often in real life because sneakers normally come in pairs. So кроссовки is very common. But if you specifically mean one sneaker, кроссовка is correct.

You may also hear colloquial кроссовок as a singular form, but кроссовка is the standard form learners should know.

Because the subject is singular: одна кроссовка.

In the present tense, Russian verbs agree with the subject in number:

• лежит = singular
• лежат = plural

So:

• Одна кроссовка лежит... = singular
• Кроссовки лежат... = plural

Unlike the past tense, the present tense does not show gender on the verb.

In Russian, the verb to be is usually omitted in the present tense.

So where English says:

• The sneaker is under the sofa

Russian normally just says:

• Кроссовка под диваном

In your sentence, there is also the verb лежит, which means something like is lying / lies. So Russian does not need a separate word for is here.

Because под uses different cases depending on the meaning:

• под + Instrumental = location, under
• под + Accusative = motion, to a position under

Here the sneaker is already there, so this is location, not movement. That is why Russian uses the instrumental:

• под диваном = under the sofa

Compare:

• Кроссовка лежит под диваном = The sneaker is lying under the sofa
• Я положил кроссовку под диван = I put the sneaker under the sofa

Диваном is the instrumental singular of диван.

The basic form is:

• диван = nominative singular

After под in a location meaning, it becomes instrumental:

• под диваном

For a regular masculine noun like диван, the instrumental singular ending is often -ом.

So:

• диван → диваном

Russian often uses different verbs depending on the physical position of something.

• лежать = to lie, to be lying
• стоять = to stand
• сидеть = to sit
• находиться = to be located, to be situated

Since a sneaker is thought of as being in a lying position, лежит sounds natural.

You could say находится instead, but that is more neutral or formal:

• Одна кроссовка находится под диваном

That is grammatical, but less everyday than лежит.

Yes. Russian word order is more flexible than English.

Your sentence is normal and natural:

• Одна кроссовка лежит под диваном.

But you could also say:

• Под диваном лежит одна кроссовка.

This puts more focus on where it is.

So the word order can change, but the emphasis changes too. The original version is a good neutral sentence for a learner.

Its basic meaning is still one, but Russian has no articles like a or the, so words like один / одна can sometimes help introduce a single item.

So одна кроссовка can feel like:

• one sneaker
• or a single sneaker

It is not a true article, though. It still keeps the idea of one more strongly than English a does.

Yes, it is possible, but the meaning changes slightly.

• Одна кроссовка лежит под диваном emphasizes that it is one sneaker
• Кроссовка лежит под диваном just states that a/the sneaker is lying under the sofa, depending on context

So одна is useful if you want to stress the number, especially if the fact that only one shoe is there matters.