Её фартук висит на кухне рядом с прихваткой и полотенцем.

Фартук is in the nominative singular. It is the subject of the sentence: it is the thing that is hanging.

This is also the dictionary form of the noun:

• фартук = apron

So the basic structure is:

• Её фартук = her apron
• висит = is hanging

Because only фартук is the subject.

The sentence mentions:

• фартук = the apron
• прихваткой и полотенцем = with / next to the potholder and towel

But прихваткой and полотенцем are not extra subjects. They are part of the phrase рядом с... meaning next to...

So the verb agrees only with фартук, which is singular:

• фартук висит = the apron is hanging

If the apron and towel were both subjects, then you would use plural:

• Фартук и полотенце висят.

Russian often uses висеть when something is in a hanging position.

So:

• висит = is hanging

This is more natural than a neutral location verb if the object is actually hanging on a hook, handle, wall, etc.

Compare:

• Фартук висит на кухне = The apron is hanging in the kitchen
• Фартук находится на кухне = The apron is located in the kitchen

The second version is grammatical, but it sounds more formal and less natural in ordinary everyday speech.

In Russian, на кухне is the normal everyday way to say in the kitchen.

This is partly idiomatic: Russian uses на with some places where English uses in.

So:

• на кухне = in the kitchen

You may sometimes see в кухне, but it is much less natural in ordinary speech and can sound more literal or stylistically marked.

Also note the case:

• на кухне uses the prepositional case for location

Compare:

• на кухне = in the kitchen, located there
• на кухню = to the kitchen, motion toward it

Кухне is in the prepositional case.

That is because на can take:

• prepositional for location
• accusative for motion toward a place

Here the apron is already there, so this is location:

• на кухне = in the kitchen

If someone were moving something there, you would use accusative:

• повесить фартук на кухню would not be natural here, but grammatically it shows motion toward a place
• more natural motion examples are things like идти на кухню = to go to the kitchen

Both nouns are in the instrumental case because they follow рядом с, which requires the instrumental.

• рядом с = next to / beside

So:

• прихватка → прихваткой
• полотенце → полотенцем

The endings look different because the nouns belong to different declension patterns:

• прихватка is a feminine noun ending in -а
• полотенце is a neuter noun ending in -е

Same case, different noun types.

Yes, in standard Russian, рядом с is followed by the instrumental case.

Examples:

• рядом с домом = next to the house
• рядом с сестрой = next to the sister
• рядом с окном = next to the window

So in your sentence:

• рядом с прихваткой и полотенцем = next to the potholder and the towel

Because one preposition can govern both nouns when they are connected by и.

So:

• рядом с прихваткой и полотенцем

means:

• next to the potholder and the towel

Russian does not need to repeat the preposition here.

You could repeat it for emphasis or clarity in some contexts, but normally you do not:

• рядом с прихваткой и с полотенцем

That is possible, but less neutral here.

Here её is a possessive pronoun meaning her.

So:

• её фартук = her apron

A useful point for learners: this possessive её does not change form in modern standard Russian.

You use the same её with different nouns and in different cases:

• её фартук
• её книга
• я вижу её фартук
• нет её фартука

The noun changes, but её stays the same.

Because свой usually refers back to the subject of the clause.

In this sentence, the subject is:

• фартук = the apron

So свой фартук would mean something like its own apron, which does not make sense here.

That is why Russian uses:

• её фартук = her apron

You use свой when the possessor is the subject:

• Она повесила свой фартук на кухне. = She hung her apron in the kitchen.

There, the subject is она, so свой is correct.

Yes. Russian word order is fairly flexible.

This sentence:

• Её фартук висит на кухне рядом с прихваткой и полотенцем.

is natural and straightforward. It starts with её фартук, which makes that the topic.

Other word orders are also possible, depending on emphasis:

• На кухне рядом с прихваткой и полотенцем висит её фартук.
• Рядом с прихваткой и полотенцем на кухне висит её фартук.

The core meaning stays the same, but the focus shifts slightly.

Yes, ё matters.

Её is pronounced roughly yi-YO.

Important points:

• ё is pronounced yo
• it is normally stressed

So:

• её sounds different from what a learner might guess from plain е

Also, in many printed texts, Russian writers omit the dots and write ее instead of её. But the intended word is often still pronounced её.

For learners, it is very helpful to remember the full form with dots:

• её = her

Because the sentence is talking about:

• one potholder
• one towel

So Russian uses singular instrumental forms:

• с прихваткой
• с полотенцем

If there were several of each, you would use plural instrumental forms instead:

• рядом с прихватками и полотенцами = next to potholders and towels

So the singular here simply matches the meaning.