Er ligt een schone vaatdoek naast de gootsteen.

Here er is the Dutch existential er. It works a lot like English there in there is / there are.

So:

• Er ligt een schone vaatdoek naast de gootsteen.
• literally: There lies a clean dishcloth next to the sink.
• natural English: There is a clean dishcloth next to the sink.

This er does not mean a specific place like there, over there. It is mainly there to introduce the existence of something.

Dutch often uses a position verb where English would simply use is.

Common ones are:

• staan = stand
• liggen = lie
• zitten = sit / be located

A vaatdoek is seen as something lying flat, so Dutch normally says:

• De vaatdoek ligt naast de gootsteen.

Using is is not the normal choice here. Dutch prefers the more specific verb liggen.

Because Dutch main clauses follow the verb-second rule.

In this sentence:

• Er is in first position
• ligt must come second
• the noun phrase een schone vaatdoek comes after that

So the structure is:

• Er
  • ligt
    • een schone vaatdoek
      • naast de gootsteen

This is very normal Dutch word order.

Because schone is an adjective placed before the noun, and in Dutch such adjectives usually take -e.

Here:

• een schone vaatdoek = a clean dishcloth

The noun vaatdoek is a de-word, so the adjective takes -e.

Compare:

• een schone vaatdoek
• de schone vaatdoek

Both have schone.

Because the noun is singular:

• een vaatdoek = one dishcloth

So the verb must also be singular:

• ik lig
• jij ligt
• hij/zij/het ligt
• er ligt een vaatdoek

If it were plural, you would use liggen:

• Er liggen schone vaatdoeken naast de gootsteen.

Because gootsteen is a de-word in Dutch.

So:

• de gootsteen = the sink

Unfortunately, whether a noun takes de or het often has to be learned with the noun itself. There is not always a simple logic you can rely on.

Naast is a preposition. It means:

• next to
• beside

So:

• naast de gootsteen = next to the sink

It directly takes its object, so you do not need an extra preposition like aan here.

Yes. That is also correct.

Compare:

• Er ligt een schone vaatdoek naast de gootsteen.
• Een schone vaatdoek ligt naast de gootsteen.

The version with er is more like introducing something new: there is a clean dishcloth...

The version without er puts een schone vaatdoek more directly in focus as the topic of the sentence.

Both are grammatical.

Because that is a different adjective position.

Dutch has two common patterns:

1. Attributive adjective: before a noun

   • een schone vaatdoek

2. Predicative adjective: after a linking verb like zijn

   • De vaatdoek is schoon

In the second pattern, the adjective usually does not get -e.

So:

• een schone vaatdoek
• de vaatdoek is schoon

It would be:

• Er liggen schone vaatdoeken naast de gootsteen.

Changes:

• ligt → liggen because the subject is plural
• vaatdoek → vaatdoeken for the plural noun
• een disappears, because English some is often left untranslated in Dutch
• schone stays schone

In this neutral there is/there are type of sentence, er is very common and usually expected.

But if you restructure the sentence, you may not need it:

• Een schone vaatdoek ligt naast de gootsteen.
• Naast de gootsteen ligt een schone vaatdoek.

So er is especially typical when you want to present something as existing or being present somewhere.

It is a prepositional phrase of location. It tells you where the dishcloth is.

So the sentence has these main parts:

• Er = existential starter
• ligt = verb
• een schone vaatdoek = the thing being talked about
• naast de gootsteen = its location

That final phrase answers the question where?