Während die Suppe auf dem Herd köchelt, lese ich das Kochbuch.

Because während is a subordinating conjunction. In German, a subordinate clause usually sends the conjugated verb to the end.

So:

• Während die Suppe auf dem Herd köchelt = While the soup is simmering on the stove

The basic pattern is:

• während + subject + other information + verb

That is why you get die Suppe ... köchelt, not die Suppe köchelt ... in that clause.

Because German main clauses follow the verb-second rule.

The whole first clause:

• Während die Suppe auf dem Herd köchelt

comes first and takes up position 1 in the sentence. That means the conjugated verb of the main clause must come next, in position 2:

• ... , lese ich das Kochbuch.

So the structure is:

1. subordinate clause
2. conjugated verb of the main clause
3. subject
4. the rest

If you put the main clause first, then you would say:

• Ich lese das Kochbuch, während die Suppe auf dem Herd köchelt.

Both are correct.

Here während means while.

It connects two actions that happen at the same time:

• the soup is simmering
• I am reading the cookbook

A useful thing to know is that während can also be used as a preposition, meaning during:

• während des Essens = during the meal

But in your sentence, it is a conjunction introducing a clause, so it means while.

Because this sentence describes a location, not movement.

German uses different cases with some prepositions depending on whether you mean:

• where something is → dative
• where to something goes → accusative

With auf:

• auf dem Herd = on the stove → location, so dative
• auf den Herd = onto the stove → movement toward it, so accusative

In your sentence, the soup is already there, so:

• auf dem Herd

is correct.

Because Herd is a masculine noun: der Herd.

After auf in a location meaning, you need the dative case. The dative form of der is dem.

So:

• der Herd = the stove
• auf dem Herd = on the stove

Kochen usually means to cook or to boil.

Köcheln means to simmer or to bubble gently, so it is more specific and softer than kochen.

Compare:

• Das Wasser kocht. = The water is boiling.
• Die Suppe köchelt. = The soup is simmering.

So köchelt is a very natural choice here, because soup often simmers rather than boils hard.

Because in German, a subordinate clause is normally separated from the main clause by a comma.

So:

• Während die Suppe auf dem Herd köchelt, lese ich das Kochbuch.

That comma is required in standard German.

Because German nouns have grammatical gender, and the article has to match the noun.

• die Suppe → Suppe is feminine
• das Kochbuch → Kochbuch is neuter

This gender is something you usually need to learn together with each noun:

• die Suppe
• das Kochbuch
• der Herd

Yes. Lese is the present tense form of lesen for ich:

• ich lese = I read / I am reading

German present tense often covers both simple present and present progressive meanings that English separates:

• Ich lese das Kochbuch. can mean
  • I read the cookbook (less likely in this context)
  • I am reading the cookbook (more likely here)

In this sentence, the meaning is clearly I am reading the cookbook while the soup is simmering.

Yes, absolutely.

That version is also correct:

• Ich lese das Kochbuch, während die Suppe auf dem Herd köchelt.

The difference is mainly in focus:

• Während die Suppe auf dem Herd köchelt, lese ich das Kochbuch.
  • starts with the time/background situation
• Ich lese das Kochbuch, während die Suppe auf dem Herd köchelt.
  • starts with what I am doing

Both are natural.

Using das Kochbuch means the cookbook — a specific cookbook that is known from the situation or context.

Using ein Kochbuch would mean a cookbook, any cookbook, or one not previously identified.

So:

• lese ich das Kochbuch = I am reading the cookbook
• lese ich ein Kochbuch = I am reading a cookbook

Both are grammatically possible, but they mean slightly different things.

If während is being used as a conjunction, yes: it introduces a subordinate clause, so the verb goes to the end.

Examples:

• Während ich koche, höre ich Musik.
• Während die Kinder schlafen, arbeiten wir.

But if während is a preposition, it is followed by a noun phrase, not a full clause:

• während des Films
• während der Pause

So the structure depends on how während is being used.