В ванной есть крючок, на котором висит полотенце.

Because after в when you mean location (not motion), Russian normally uses the prepositional case.

So:

• в ванной = in the bathroom
• ванной is the prepositional singular form

Also, ванная here is a shortened noun-like form of ванная комната meaning bathroom. So в ванной is really a shortened way of saying в ванной комнате.

That is a very common question. Russian usually does leave out to be in sentences like:

• Это крючок = This is a hook
• Полотенце мокрое = The towel is wet

But есть is often used in existential sentences, where the idea is there is / there are.

So:

• В ванной есть крючок = There is a hook in the bathroom

Here, есть does not function like the normal English is. It signals existence/presence.

A useful contrast:

• Это крючок = This is a hook
• В ванной есть крючок = There is a hook in the bathroom

Because in a sentence with есть, the thing that exists is normally in the nominative.

So in:

• В ванной есть крючок

the noun крючок is the thing being introduced, so it stays in nominative singular.

A very useful comparison is the negative version:

• В ванной нет крючка = There is no hook in the bathroom

With нет, Russian usually uses the genitive, so крючок becomes крючка.

That nominative/genitive contrast is worth remembering:

• есть крючок
• нет крючка

На котором means on which or that ... on.

It comes from the relative pronoun который = which / that.

In this sentence, it refers back to крючок, so it has to agree with крючок:

• крючок is masculine
• singular
• so the pronoun is based on masculine singular который

But because of the preposition на, the pronoun must be in the prepositional case:

• который → котором

So:

• на котором висит полотенце = on which a towel is hanging

Because the relationship between the towel and the hook is specifically on the hook.

Russian has to show that explicitly:

• на котором висит полотенце = on which the towel is hanging

If you used just который, the grammar would be incomplete for this meaning, because висеть here needs the idea of location/support: the towel is hanging on something.

So на is not optional; it is part of the meaning.

Because its subject is полотенце, which is singular.

So the structure is:

• полотенце висит = the towel hangs / is hanging

Since полотенце is singular, the verb is singular too:

• висит

If there were multiple towels, you would use plural:

• полотенца висят = the towels are hanging

Because полотенце is the subject of висит, not a direct object.

In English, learners sometimes expect something like the hook holds the towel, but that is not what the Russian sentence says. The Russian structure is closer to:

• On the hook, a towel is hanging

So:

• полотенце = the thing doing the hanging
• therefore it is nominative

That is why you get:

• висит полотенце

not an accusative form.

Russian often prefers more specific position/state verbs where English might simply use is.

Some very common ones are:

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

So Russian naturally says:

• На столе лежит книга = There is a book on the table
• У двери стоит стул = There is a chair by the door
• На крючке висит полотенце = There is a towel hanging on the hook

Using висит makes the image more natural in Russian.

Because на котором висит полотенце is a relative clause describing крючок.

Russian normally separates this kind of clause with a comma:

• крючок, на котором висит полотенце

This is standard punctuation with clauses introduced by forms of который.

So the comma is not optional here in normal writing.

Yes, you could say:

• В ванной крючок, на котором висит полотенце.

But that sounds a bit less neutral and less like a straightforward there is sentence.

With есть, the sentence clearly introduces something as existing/present:

• В ванной есть крючок...

That is the safest and most natural version for learners to use when they want to say there is.

Yes. That version is perfectly natural.

It means the same thing, but the structure is different:

• В ванной есть крючок, на котором висит полотенце.
  One sentence with a relative clause: a hook on which a towel is hanging

• В ванной есть крючок, на нём висит полотенце.
  Two clauses: There is a hook in the bathroom; a towel is hanging on it

The на котором version sounds a bit more compact and connected. The на нём version can sound slightly more conversational.

It is the normal everyday word for a small hook of the kind you would hang a towel on.

Historically, the ending -ок can feel diminutive in some words, but in this case крючок is simply the standard practical word for a hook like this.

So in a bathroom context, крючок is exactly what a Russian speaker would expect.