Войлочная шапка и банная простыня лежат на верхнем полке.

Because the subject contains two items joined by и:

• войлочная шапка
• банная простыня

Together they form a plural subject, so the verb is plural: лежат.

Compare:

• Шапка лежит на полке. = The hat is lying on the shelf.
• Шапка и простыня лежат на полке. = The hat and the sheet are lying on the shelf.

They are adjectives, and they must agree with the nouns they describe in gender, number, and case.

Here:

• шапка is feminine singular nominative
• простыня is feminine singular nominative

So the adjectives also take feminine singular nominative endings:

• войлочная шапка
• банная простыня

Very roughly:

• masculine: войлочный
• feminine: войлочная
• neuter: войлочное
• plural: войлочные

No — in standard Russian, this is not correct.

It should be:

на верхней полке

Why? Because полка is a feminine noun, and the adjective must agree with it.

• nominative: верхняя полка
• prepositional: на верхней полке

The form верхнем is masculine/neuter prepositional, so it does not match полке.

So the corrected sentence is:

Войлочная шапка и банная простыня лежат на верхней полке.

Полке is in the prepositional case.

After на, when you are talking about location rather than motion, Russian normally uses the prepositional case:

• на полке = on the shelf
• на столе = on the table
• на верхней полке = on the upper shelf

But if you are talking about motion onto something, Russian uses the accusative:

• положить на полку = to put onto the shelf
• положить на верхнюю полку = to put onto the upper shelf

So:

• лежат на верхней полке = are lying on the upper shelf
• положить на верхнюю полку = put onto the upper shelf

Russian often uses a more specific verb for the position of an object.

лежать means to lie / be lying and is commonly used for things that are resting somewhere, especially in a horizontal position.

So Russian prefers:

• Книга лежит на столе.
• Шапка лежит на полке.

rather than using a simple equivalent of English is.

In English, The hat and the towel are on the upper shelf sounds perfectly normal. In Russian, лежат sounds natural because it describes how the objects are positioned.

That is a good question, because простыня usually does mean bed sheet.

But банная простыня can mean a bath sheet — a large towel-like cloth used after bathing or in a sauna/bathhouse context.

So depending on context, it may refer to:

• a literal sheet used in a bathhouse
• a very large bath towel / bath sheet

This is one of those cases where the exact English translation depends on context, even though the Russian wording is normal.

Because many Russian feminine nouns end in -я as well as -а.

So простыня is feminine, just like:

• земля
• неделя
• семья

That is why adjectives agreeing with it are feminine too:

• банная простыня
• белая простыня
• чистая простыня

So the ending -я does not make a noun masculine or neuter; many -я nouns are feminine.

Russian word order is fairly flexible.

The version here is close to a neutral pattern:

Войлочная шапка и банная простыня лежат на верхней полке.

But you could also say:

На верхней полке лежат войлочная шапка и банная простыня.

This version emphasizes the location first: On the upper shelf, there are...

Both are natural. The difference is mostly one of emphasis, not basic meaning.

The usual stresses are:

• войло́чная
• ша́пка
• ба́нная
• простыня́
• лежа́т
• на ве́рхней по́лке

So the corrected full sentence is pronounced:

Войло́чная ша́пка и ба́нная простыня́ лежа́т на ве́рхней по́лке.