На кухне рядом со столом стоит деревянная табуретка.

Because на here means in/on in the sense of location, and after на to express location, Russian usually uses the prepositional case.

• кухня = kitchen
• на кухне = in the kitchen

So:

• на кухню = to the kitchen (movement, accusative)
• на кухне = in the kitchen (location, prepositional)

Even though English says in the kitchen, Russian commonly uses на кухне, not в кухне.

Рядом means near / nearby, but to say next to something, Russian uses the fixed expression:

рядом с + instrumental

So:

• рядом с домом = next to the house
• рядом с окном = next to the window
• рядом со столом = next to the table

The form со is just a pronunciation variant of с. Russian often uses со instead of с when it sounds easier before certain consonant combinations.

So here:

• с столом would sound awkward
• со столом is the normal form

Because after рядом с, the noun goes in the instrumental case.

• стол = table
• со столом = next to the table

This is just something you learn together with the phrase:

рядом с + instrumental

Other examples:

• рядом с братом = next to my brother
• рядом с машиной = next to the car
• рядом со столом = next to the table

Russian often uses verbs like стоять, лежать, and висеть to describe where things are, depending on their physical position.

• стоять = to stand
• лежать = to lie
• висеть = to hang

A stool is thought of as something that stands, so:

стоит деревянная табуретка = a wooden stool is standing / there is a wooden stool standing

This sounds very natural in Russian.

You could also say:

• На кухне рядом со столом находится деревянная табуретка.

But находится is more formal, less everyday, and often less vivid.

Russian usually does not use есть the way English uses there is in simple location sentences like this.

Because табуретка is a feminine singular noun in the nominative case, and the adjective must agree with it.

• деревянный = wooden
• feminine nominative singular: деревянная
• табуретка = stool

So:

• деревянный стол = wooden table
• деревянная табуретка = wooden stool
• деревянное окно = wooden window

The adjective matches the noun in gender, number, and case.

Because it is the subject of the sentence.

The basic idea is:

What is standing?
Деревянная табуретка.

So табуретка is nominative, and the adjective деревянная also stays nominative to agree with it.

Russian word order is flexible, and different orders can change the focus or information structure.

This sentence starts with the place:

• На кухне = in the kitchen
• рядом со столом = next to the table

Then it gives the main new information:

• стоит деревянная табуретка = there is a wooden stool standing

So this order feels natural if you are first setting the scene and then saying what is there.

You could also say:

Деревянная табуретка стоит на кухне рядом со столом.

That is also grammatical, but it sounds a bit more like you are talking about the stool specifically and then saying where it is.

Not exactly.

• стул = chair
• табуретка = stool

A табуретка usually has no backrest, while a стул usually does.

So in this sentence, табуретка is specifically a stool, not just any chair.

No. Russian does not have articles like a/an and the.

So:

• табуретка can mean a stool or the stool
• стол can mean a table or the table

Which one is meant depends on context.

In this sentence, English might translate it as either:

• There is a wooden stool next to the table in the kitchen
• The wooden stool is standing next to the table in the kitchen

Usually, without context, English speakers often choose a wooden stool.

It can be used both ways, depending on what you want to say.

1. By itself:
   рядом = nearby / close by

• Он живёт рядом. = He lives nearby.

2. With a noun:
   рядом с + instrumental = next to / near

• Он живёт рядом с парком. = He lives near the park.
• Табуретка стоит рядом со столом. = The stool is next to the table.

So in your sentence, since a specific object is named, Russian uses рядом со столом.

Yes. That would also be natural.

• рядом со столом = next to / near the table
• возле стола = near/by the table
• около стола = near the table

But the grammar changes:

• рядом с + instrumental → со столом
• возле + genitive → стола
• около + genitive → стола

So:

• На кухне рядом со столом стоит деревянная табуретка.
• На кухне возле стола стоит деревянная табуретка.

Both are fine, with slightly different shades of style and emphasis.

Yes. The verb agrees with the subject.

Here the subject is singular:

• табуретка = one stool
• so: стоит = stands

If the subject were plural:

• На кухне рядом со столом стоят деревянные табуретки.
• There are wooden stools standing next to the table in the kitchen.

So:

• стоит = singular
• стоят = plural

The main stresses are:

• на ку́хне
• ря́дом
• со столо́м
• стои́т
• деревя́нная
• табуре́тка

Stress is important in Russian because it is not always predictable. A learner may want to practice the whole sentence like this:

На ку́хне ря́дом со столо́м стои́т деревя́нная табуре́тка.

Yes, that is a very close literal translation, and it helps show how Russian works.

But in natural English, people often say:

• There is a wooden stool in the kitchen next to the table.

Russian often uses a normal position verb like стоит where English prefers there is.

So the literal translation is useful for understanding the grammar, but the most natural English translation may sound a little different.