U kupatilu su peškir i sapun.

Because u can take different cases depending on the meaning:

• u + accusative = movement into something
• u + locative = being in something

In this sentence, the meaning is location, not motion, so Serbian uses the locative:

• kupatilo = bathroom
• u kupatilu = in the bathroom

Compare:

• Idem u kupatilo. = I am going into the bathroom.
• Ja sam u kupatilu. = I am in the bathroom.

Because together they form a compound subject: peškir i sapun = the towel and the soap.

Even though each noun is singular on its own, the whole subject is plural, so the verb must also be plural:

• peškir je = the towel is
• sapun je = the soap is
• peškir i sapun su = the towel and the soap are

So su is correct because there are two things.

Su is the normal short form, or clitic, of jesu.

In everyday Serbian, short forms are much more common:

• oni jesu = they are / they do indeed are
• oni su = they are

In this sentence, su is the natural choice.
Jesu would sound more emphatic or contrastive, for example if you were correcting someone:

• Nisu u kuhinji, nego jesu u kupatilu. = They are not in the kitchen, but they are in the bathroom.

Because short forms like su usually go in second position in the sentence or clause. This is a very important Serbian word-order rule.

Here, the first unit is U kupatilu, and the clitic su comes right after that:

• U kupatilu su peškir i sapun.

This is normal Serbian word order.

You could also say:

• Peškir i sapun su u kupatilu.

That is also correct, but the focus is a little different.
The original sentence emphasizes the location first: In the bathroom...

They are in the nominative case, because they are the subject of the sentence.

The subject answers the question what is in the bathroom?

• peškir = towel
• sapun = soap

Since they are the things that are somewhere, they stay in nominative:

• Peškir je ovde. = The towel is here.
• Sapun je tamo. = The soap is there.

In U kupatilu su peškir i sapun, both nouns are subjects, so both are nominative.

Yes, the structure is closer to In the bathroom are a towel and soap than to English There is/are.

Serbian usually does not need a dummy subject like there. It simply states the location and the things الموجود? Wait must be English only. Need fix. Let's continue.

Serbian simply states the location and the things that are there:

• Na stolu je knjiga. = There is a book on the table.
• U sobi su dva prozora. = There are two windows in the room.

So Serbian expresses existence/location without a special word like English there.

Because Serbian does not have articles like English a/an and the.

So:

• peškir can mean a towel or the towel
• sapun can mean a soap, soap, or the soap, depending on context

The exact interpretation comes from the situation, context, or word order.

In a sentence like U kupatilu su peškir i sapun, English usually translates it naturally as something like:

• There are a towel and soap in the bathroom or more naturally
• In the bathroom there are a towel and some soap or, depending on context,
• The towel and the soap are in the bathroom

Serbian leaves that open unless the context makes it specific.

Yes. That is completely correct.

Both sentences are grammatical:

• U kupatilu su peškir i sapun.
• Peškir i sapun su u kupatilu.

The difference is mainly focus:

• U kupatilu su peškir i sapun.
  Focuses first on where the items are.
• Peškir i sapun su u kupatilu.
  Focuses first on what the items are.

So the original sentence sounds especially natural if someone is talking about what is in the bathroom.

Normally, no. The standard form is su, because the subject is plural overall:

• peškir i sapun = two things
• therefore: su

Using je here would sound ungrammatical to most speakers in standard Serbian.

Both peškir and sapun are masculine singular nouns.

That matters mainly when they are used individually with adjectives, pronouns, or past-tense forms. For example:

• nov peškir = a new towel
• dobar sapun = a good soap

But in this sentence, the important agreement is with the whole compound subject, which is plural:

• peškir i sapun su

So even though each noun is masculine singular, together they trigger the plural verb su.

Yes. I means and.

So:

• peškir i sapun = towel and soap

It joins the two nouns into one compound subject. Because of that, the verb becomes plural:

• peškir i sapun su

It can mean either, depending on context, because Serbian has no articles.

So u kupatilu could be:

• in the bathroom
• in a bathroom

Context tells you which one is meant. If the speaker and listener already know which bathroom, English often uses the bathroom. If it is not specific, English might use a bathroom.

A very natural English rendering would often be:

• The towel and the soap are in the bathroom. or
• There are a towel and some soap in the bathroom.

Serbian is more flexible with word order than English, because case endings and agreement already show the grammatical relationships.

So Serbian can move elements around for emphasis much more easily:

• U kupatilu su peškir i sapun.
• Peškir i sapun su u kupatilu.

Both are correct; the choice depends on what the speaker wants to emphasize.