Na het eten maak ik de tafel schoon met een natte vaatdoek.

Here eten is being used like a noun: the eating / the meal. After na you need a noun phrase, so na het eten means after the meal or after eating.

Dutch often uses an infinitive as a noun with het, for example:

• het eten = the food / the meal / eating
• het lezen = reading
• het werken = working

So na het eten is a very natural fixed expression.

Because Dutch main clauses usually follow the verb-second rule.

If something other than the subject comes first, the finite verb must still stay in the second position. So:

• Ik maak de tafel schoon.
• Na het eten maak ik de tafel schoon.

In the second sentence, Na het eten takes the first position, so maak comes next, and ik follows it.

This is one of the most important word-order patterns in Dutch.

Because schoonmaken is treated as a separable verb in this kind of sentence.

In a main clause, the finite verb part goes to the usual verb position, and the other part stays later in the sentence:

• Ik maak de tafel schoon.
• Wij maken de keuken schoon.

But when the verb stays together, it is written as one word:

• Ik wil de tafel schoonmaken.
• Omdat ik de tafel schoonmaak, is hij straks klaar.

So maak ... schoon and schoonmaken are two forms of the same verb pattern.

With schoonmaken, the thing being cleaned comes before schoon:

• iets schoonmaken
• de tafel schoonmaken
• het raam schoonmaken

So in the sentence:

• maak ik de tafel schoon

de tafel is the direct object, and schoon describes the result: you make the table clean.

English learners sometimes want to move schoon earlier, but in Dutch this pattern is normal and fixed.

Because tafel is a de-word in Dutch: de tafel.

Dutch nouns have grammatical gender classes, and singular nouns usually take either de or het. Unfortunately, this usually has to be learned with the noun itself.

So you learn:

• de tafel
• de stoel
• het raam
• het huis

There is no special reason in this sentence; it is simply the correct article for tafel.

Because Dutch adjectives usually take -e before a noun, and that happens here.

You have:

• een natte vaatdoek

This is because vaatdoek is a de-word: de vaatdoek. With a singular de-word and an indefinite article een, the adjective normally gets -e.

Compare:

• een natte vaatdoek = a wet dishcloth
• de natte vaatdoek = the wet dishcloth

But with some het-words, the indefinite singular adjective may stay without -e:

• een nat glas
• het natte glas

So natte here is completely regular.

Yes, it is one word. Dutch very often forms compound nouns by joining words together.

vaatdoek is made from:

• vaat = dishes / washing-up
• doek = cloth

So literally it is something like dish cloth, and in normal English you would say dishcloth or dish rag, depending on context.

Also, the last part usually determines the article, and doek is a de-word, so:

• de vaatdoek

Met means with, and here it introduces the instrument used to do the action.

So:

• met een natte vaatdoek = using a wet dishcloth

This is a very common use of met in Dutch:

• Ik schrijf met een pen.
• Hij opent de deur met een sleutel.
• Zij maakt de tafel schoon met een doek.

So met here tells you what the speaker uses to clean the table.

Yes. Dutch allows some flexibility, although the word order changes according to the usual rules.

For example:

• Na het eten maak ik de tafel schoon.
• Ik maak de tafel na het eten schoon.

Both are possible. Putting Na het eten first gives it a bit more prominence, as if setting the time frame first.

What you cannot do is ignore the verb-second rule in a main clause. So after fronting Na het eten, you need:

• Na het eten maak ik ...

not:

• Na het eten ik maak ...

Sometimes, but not always.

Schoonmaken is the general verb for to clean. It works very naturally for tables, rooms, kitchens, bathrooms, and many everyday cleaning situations.

Poetsen is also used for cleaning, but it often suggests polishing, brushing, or a more specific kind of cleaning. For example:

• tanden poetsen = brush teeth
• schoenen poetsen = polish shoes

For a table after eating, de tafel schoonmaken is the most neutral and natural choice.