Ich wische die Krümel mit der Küchenrolle weg.

Because the verb here is wegwischen, which is a separable verb.

In a normal main clause, German splits separable verbs:

• Ich wische die Krümel mit der Küchenrolle weg.

The conjugated part goes in second position, and the prefix goes to the end.

You see the full verb together again in forms like:

• infinitive: wegwischen
• past participle: weggewischt

So this sentence is built from wegwischen = to wipe away.

wischen means to wipe.

Adding weg- gives the idea of away, off, or out of the way:

• wischen = to wipe
• wegwischen = to wipe away

So:

• Ich wische die Krümel mit der Küchenrolle. = I wipe the crumbs with the paper towel.
• Ich wische die Krümel mit der Küchenrolle weg. = I wipe the crumbs away.

The version with weg makes it clearer that the crumbs are being removed.

It is accusative.

The noun Krümel is plural here:

• singular: der Krümel = the crumb
• plural: die Krümel = the crumbs

In German, the plural definite article is die in both the nominative and the accusative:

• nominative plural: die Krümel
• accusative plural: die Krümel

So nothing changes in the article here, even though die Krümel is the direct object.

Because mit always takes the dative case.

The noun Küchenrolle is feminine:

• nominative: die Küchenrolle
• dative: der Küchenrolle

So:

• mit der Küchenrolle = with the paper towel / kitchen roll

This is a very common pattern:

• mit dem Messer
• mit der Hand
• mit den Fingern

die Küchenrolle literally means kitchen roll, and in everyday usage it usually refers to paper towel or a roll of paper towels.

Depending on context, English might translate it as:

• paper towel
• kitchen roll
• paper towels

In this sentence, it means the thing being used to wipe up the crumbs.

A learner should just remember that Küchenrolle is the normal household paper towel item, even if English words it a bit differently.

German main clauses normally put the conjugated verb in second position:

• Ich wische ...

Then the other parts of the sentence follow, and the separable prefix goes to the end:

• Ich wische die Krümel mit der Küchenrolle weg.

A simple breakdown is:

• Ich = subject
• wische = conjugated verb
• die Krümel = direct object
• mit der Küchenrolle = prepositional phrase
• weg = separable prefix

That order is natural and neutral.

Yes. German word order is somewhat flexible.

You can also say:

• Ich wische mit der Küchenrolle die Krümel weg.

That is still grammatical. The difference is mostly about focus or emphasis.

• Ich wische die Krümel mit der Küchenrolle weg.
  Slightly more neutral: first mention what is being wiped away.

• Ich wische mit der Küchenrolle die Krümel weg.
  Puts a bit more attention on the tool being used.

So both are possible.

Yes, but the meaning changes slightly.

• Ich wische die Krümel mit der Küchenrolle weg.
  = I wipe the crumbs away.

• Ich wische die Krümel mit der Küchenrolle.
  = I wipe the crumbs with the paper towel.

Without weg, the sentence does not explicitly highlight that the crumbs are being removed. It may still be understood from context, but weg makes that idea clearer.

Both are possible, but they mean slightly different things.

• mit der Küchenrolle suggests a specific, known paper towel / kitchen roll, often the one at hand.
• mit einer Küchenrolle means with a paper towel / with a kitchen roll, less specific.

German often uses the definite article in everyday situations where English might be more flexible. If the speaker is talking about the paper towel they are using right now, mit der Küchenrolle sounds very natural.

The dictionary form is wegwischen.

If you look it up, you would usually find:

• wegwischen = to wipe away

In the sentence, it appears as:

• ich wische ... weg

That is just the normal present-tense form of the separable verb.