Meine Schwester vermischt Eier und Milch mit dem Schneebesen, während ich den Tisch decke.

Because meine Schwester is the subject of the main clause — the person doing the action.

The basic pattern is:

• Meine Schwester = subject
• vermischt = verb
• Eier und Milch = direct object
• mit dem Schneebesen = prepositional phrase

In the nominative case, meine Schwester is correct.
You would use meiner Schwester in other cases, for example:

• Ich helfe meiner Schwester. = I help my sister.
  Here meiner Schwester is dative.

vermischt is the 3rd person singular form of the verb vermischen, which means to mix or to blend.

The subject is meine Schwester, which is she, so the verb must match that subject:

• ich vermische
• du vermischst
• er/sie/es vermischt

So Meine Schwester vermischt ... means My sister mixes ...

In German, just like in English, you do not always need an article with nouns.

Here:

• Eier = eggs
• Milch = milk

They are being talked about in a general, ingredient-like sense, not as specific individually identified items. That is very natural in German.

Compare:

• Sie vermischt Eier und Milch. = She mixes eggs and milk.
• Sie vermischt die Eier und die Milch. = She mixes the eggs and the milk.
  This sounds more specific, as if both are already known in the situation.

Also, Milch is often used without an article when speaking about the substance in general, much like milk in English.

The preposition mit always takes the dative case.

The noun is:

• der Schneebesen = the whisk

Since Schneebesen is masculine, its dative singular form with the article is:

• dem Schneebesen

So:

• mit dem Schneebesen = with the whisk

The other forms would mean different things or be incorrect here:

• mit den Schneebesen = with the whisks (plural)
• mit der Schneebesen = incorrect, because Schneebesen is not feminine

Der Schneebesen means whisk, the kitchen tool used for beating or mixing ingredients.

It is a compound noun:

• Schnee = snow
• Besen = broom

Historically, the image is something like a tool that makes things light and fluffy. But in modern German, you should simply learn Schneebesen as the normal word for whisk.

Because während introduces a subordinate clause.

German normally separates a subordinate clause from the main clause with a comma:

• Meine Schwester vermischt Eier und Milch mit dem Schneebesen, während ich den Tisch decke.

That comma is required.

Because während is a subordinating conjunction. Subordinating conjunctions send the conjugated verb to the end of the clause.

So:

• main clause: Ich decke den Tisch.
• subordinate clause after während: während ich den Tisch decke

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

Other common conjunctions that do the same thing are:

• weil = because
• dass = that
• wenn = if/when
• obwohl = although

In a normal main clause, German usually puts the conjugated verb in second position:

• Ich decke den Tisch.

But after während, the clause becomes subordinate, so the verb moves to the end:

• während ich den Tisch decke

So the change is caused by subordinate word order, not by the verb decken itself.

Literally, decken can mean to cover, but the expression den Tisch decken is an idiom meaning to set the table.

So you should learn it as a fixed expression:

• den Tisch decken = to set the table

This is very common German.

Because den Tisch is the direct object of decken.

The noun is:

• der Tisch = the table

In the accusative case, masculine der changes to den:

• nominative: der Tisch
• accusative: den Tisch

So:

• Ich decke den Tisch. = I set the table.

Here während means while, introducing something happening at the same time:

• während ich den Tisch decke = while I set the table

It can also mean during when used as a preposition:

• während des Essens = during the meal

So it has two common uses:

1. Conjunction
   • clause
     
     • während ich den Tisch decke
2. Preposition
   • noun phrase
     
     • während des Abends

In your sentence, it is a conjunction.

German usually uses the simple present tense for actions happening now.

So:

• Meine Schwester vermischt ...
• ich decke ...

can naturally describe actions currently in progress.

English often uses the present progressive:

• My sister is mixing ... while I am setting ...

German normally does not need a special progressive form here.

Yes, absolutely.

You can say:

• Während ich den Tisch decke, vermischt meine Schwester Eier und Milch mit dem Schneebesen.

That is completely natural.

Notice what happens in the main clause after the subordinate clause:

• The verb vermischt comes immediately after the opening subordinate clause.

This is because German main clauses still keep the verb in second position, and the whole subordinate clause counts as the first element.

No. German has several verbs related to mixing, and they are not always exactly the same.

For example:

• vermischen = to mix together, combine
• mischen = to mix
• verrühren = to stir together thoroughly
• schlagen = to beat, whisk

So in a kitchen context, depending on the exact action, a German speaker might also say:

• Meine Schwester verrührt Eier und Milch ...
• Meine Schwester schlägt Eier und Milch mit dem Schneebesen ...

But vermischt is perfectly correct and understandable.

The standard order here is very natural:

• Meine Schwester vermischt Eier und Milch mit dem Schneebesen

But German word order is somewhat flexible, especially with adverbial and prepositional phrases. For example, you could also hear:

• Meine Schwester vermischt mit dem Schneebesen Eier und Milch.

That is possible, though the original version sounds more neutral and natural in many contexts.

What is much less flexible is the position of the conjugated verb in a main clause: it must stay in second position.

So this part is fixed:

• Meine Schwester vermischt ...

After that, the order of the other elements can vary more.