Ég set viskustykkið í vaskinn eftir kvöldmat.

Because setja is the infinitive, meaning to put / to place. After ég (I), Icelandic uses the conjugated verb form:

• að setja = to put
• ég set = I put
• þú setur = you put
• hann/hún/það setur = he/she/it puts

So ég set is the correct present-tense form.

Viskustykkið is the direct object — it is the thing being put somewhere.

The basic noun is:

• viskustykki = dishcloth / tea towel

The ending -ið makes it definite:

• viskustykki = a dishcloth
• viskustykkið = the dishcloth

So this word means the dishcloth.

It is functioning as the direct object of setja, so it is in the accusative.

A useful detail: viskustykki is a neuter noun, and in many neuter nouns the nominative and accusative singular look the same. So even though the form is viskustykkið, its job in this sentence is accusative because it is the object of the verb.

Because í can take different cases depending on meaning:

• accusative = motion into
• dative = location in

Here the dishcloth is being moved into the sink, so Icelandic uses the accusative:

• í vaskinn = into the sink

If the dishcloth were already there, you would use the dative:

• í vaskinum = in the sink

This is a very common Icelandic pattern with places and movement.

It means the sink.

The basic noun is:

• vaskur = sink

Here it appears as vaskinn, which is the definite accusative singular form. That fits because:

1. it is after í with movement, so accusative is needed
2. the meaning is the sink, not just a sink

In Icelandic, the definite article is usually attached to the end of the noun instead of being a separate word.

So:

• viskustykki = dishcloth
• viskustykkið = the dishcloth

and

• vaskur / vask = sink
• vaskinn = the sink

This is one of the first big differences English speakers notice.

Because kvöldmat here is being used in a general sense, like English after dinner.

In English, we often say:

• after dinner
• before breakfast

without the, and Icelandic often does the same with meals:

• eftir kvöldmat = after dinner

If you wanted to refer to a specific dinner, kvöldmatinn could be possible:

• eftir kvöldmatinn = after the dinner / after that particular evening meal

It is accusative singular because eftir in the meaning after takes the accusative.

The dictionary form is:

• kvöldmatur = dinner, evening meal

But masculine nouns like this often change in the accusative singular:

• kvöldmatur → kvöldmat

So:

• eftir kvöldmat = after dinner

Yes. Icelandic word order is flexible, but in main clauses the finite verb usually stays in second position.

The original sentence is:

• Ég set viskustykkið í vaskinn eftir kvöldmat.

You can also move the time phrase to the front:

• Eftir kvöldmat set ég viskustykkið í vaskinn.

Notice that when eftir kvöldmat comes first, set still stays in second position, before ég. That is a classic Icelandic V2 pattern.

It can often mean either, depending on context.

• Ég set viskustykkið í vaskinn eftir kvöldmat. can mean
  I put the dishcloth in the sink after dinner as a habitual action
• In the right context, it could also refer to a present or near-present action

Very often, without extra context, learners understand it as a simple present / habitual statement.

The nouns are:

• viskustykki — neuter
• vaskur — masculine
• kvöldmatur — masculine

Gender matters because it affects:

• the form of the definite article
• the case endings
• adjectives that might agree with the noun

For example, compare:

• viskustykkið — neuter definite form
• vaskinn — masculine definite accusative form

So gender is not just a dictionary detail; it changes the forms you actually use.

Sometimes, but setja is the most general verb here for put/place.

Very roughly:

• setja = put, place, set
• leggja = lay, put down, place in a lying position

In many everyday contexts, setja works well as a general put verb. If you use leggja, it may suggest more specifically laying something down. Learners are usually safest with setja unless they know the nuance they want.

A rough English-friendly approximation is:

YEG set VIS-ku-stik-kith ee VAS-kinn EFF-tir KVELD-maht

Very approximate word-by-word:

• Ég ≈ yeg
• set ≈ set
• viskustykkið ≈ VIS-ku-stik-kith
• í ≈ ee
• vaskinn ≈ VAS-kinn
• eftir ≈ EFF-tir
• kvöldmat ≈ KVELD-maht

This is only an approximation, but it can help you get started before listening to native pronunciation.