Nach dem Essen wische ich den Tisch mit dem Tuch ab.

Because Essen is being used as a noun here, not as the verb essen.

In German, verbs can often be turned into nouns. This is happening in nach dem Essen, which means something like:

• after the meal
• after eating

Since all nouns are capitalized in German, it becomes Essen with a capital E.

Because the preposition nach takes the dative case.

So:

• nominative: das Essen
• dative: dem Essen

That is why you get:

• nach dem Essen = after the meal / after eating

This is something you often just have to learn together with the preposition:

• nach + dative
• mit + dative

It means after the meal or after eating.

In this sentence, Essen can be understood in a fairly natural way as either:

• the act of eating
• the meal

So Nach dem Essen is a very common German expression meaning:

• after lunch/dinner/the meal
• once eating is finished

It functions as a time expression at the beginning of the sentence.

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

The conjugated verb here is wische, and it must be in the second position. If you put a time phrase first, then the subject usually comes after the verb.

So the structure is:

• Nach dem Essen = first element
• wische = second element
• ich = subject, after the verb

Compare:

• Ich wische nach dem Essen den Tisch mit dem Tuch ab.
• Nach dem Essen wische ich den Tisch mit dem Tuch ab.

Both are possible. The second version emphasizes the time phrase a bit more.

Because den Tisch is the direct object of the verb, so it is in the accusative case.

The verb abwischen answers the question what is being wiped?

• I wipe the table.

For the masculine noun der Tisch, the accusative form is:

• nominative: der Tisch
• accusative: den Tisch

So:

• Ich wische den Tisch ab. = I wipe the table off/down.

Because mit always takes the dative case.

So:

• nominative: das Tuch
• dative: dem Tuch

That gives:

• mit dem Tuch = with the cloth

This phrase tells you the instrument used to do the action.

A very useful rule to remember:

• mit + dative

The full verb is abwischen, which is a separable verb.

Its parts are:

• ab- = separable prefix
• wischen = base verb

In a normal main clause, the prefix separates and goes to the end:

• Ich wische den Tisch ab.

In the infinitive, they stay together:

• den Tisch abwischen

So in your sentence:

• wische is the conjugated part
• ab is the separated prefix at the end

Because abwischen is a separable-prefix verb, and in a main clause the prefix is sent to the end.

So:

• infinitive: abwischen
• main clause: Ich wische ... ab.

This happens with many German verbs, for example:

• aufstehen → Ich stehe um 7 Uhr auf.
• anrufen → Ich rufe dich morgen an.
• mitbringen → Ich bringe das Buch mit.

So ab at the end is completely normal German word order.

A useful rough distinction is:

• wischen = to wipe
• abwischen = to wipe off / wipe down

abwischen often sounds more complete and more like removing something from a surface.

For example:

• den Tisch wischen = wipe the table
• den Tisch abwischen = wipe the table down / wipe something off the table surface

In everyday German, abwischen is very natural for cleaning a table.

Yes, German word order is somewhat flexible, but the main rules still apply:

1. The conjugated verb stays in second position in a main clause.
2. The separable prefix ab stays at the end.

So these are all possible:

• Nach dem Essen wische ich den Tisch mit dem Tuch ab.
• Ich wische nach dem Essen den Tisch mit dem Tuch ab.
• Den Tisch wische ich nach dem Essen mit dem Tuch ab.

They do not all have exactly the same emphasis, though.

For example:

• Nach dem Essen... emphasizes when
• Den Tisch... emphasizes what is being wiped

The version you were given is a very natural, neutral sentence.

Literally, it means with the cloth, because dem is the dative form of das.

So the sentence refers to a specific cloth or at least a cloth understood from the situation.

If you wanted to say with a cloth, you would normally say:

• mit einem Tuch

So:

• mit dem Tuch = with the cloth
• mit einem Tuch = with a cloth

In real-life translation, English may still simply say with a cloth, depending on context.

Yes. wische is present tense.

German present tense can mean:

• something happening now
• something that happens regularly

So this sentence could mean either:

• After the meal, I am wiping the table with the cloth.
• After meals, I wipe the table with the cloth.

Usually the context tells you which meaning is intended. In many everyday situations, it sounds like a regular or typical action.