После долгой прогулки мы сидели у камина и грелись, пока чайник кипел.

Because после is a preposition that requires the genitive case.

So:

• долгая прогулка = a long walk in the dictionary/basic form
• after после, it becomes долгой прогулки = after a long walk

This is a very common pattern:

• после урока = after the lesson
• после работы = after work
• после дождя = after the rain

So the whole phrase после долгой прогулки is simply after a long walk, with both words changed into the genitive.

Because adjectives in Russian must agree with the nouns they describe.

Here:

• прогулка is a feminine singular noun
• after после, it goes into the genitive singular
• the adjective долгая must match that noun in gender, number, and case

So:

• nominative: долгая прогулка
• genitive: долгой прогулки

This agreement happens throughout Russian:

• интересная книга → интересной книги
• новая машина → новой машины

In the past tense, Russian verbs agree with the subject in gender and number.

The infinitive is сидеть = to sit.
Its past-tense forms are:

• сидел = he sat / was sitting
• сидела = she sat / was sitting
• сидело = it sat / was sitting
• сидели = they sat / were sitting

Since the subject is мы = we, the verb must be plural, so you get сидели.

Грелись comes from греться, which means to warm oneself, to get warm, or to be warming oneself.

The ending -ся / -сь marks a reflexive verb. In many cases, that means the action somehow comes back to the subject.

So:

• греть = to warm something
• греться = to warm oneself / to get warm

Compare:

• Мы грели чайник. = We were heating the kettle.
• Мы грелись у камина. = We were warming ourselves by the fireplace.

In this sentence, грелись is used because we are the ones getting warm.

Because the sentence describes an ongoing situation in the past, not a completed result.

• грелись is imperfective: it shows a continuing process
• согрелись would mean got warm / became warm and focuses on the result
• погрелись often means warmed ourselves for a while

Here the picture is: after the walk, we were sitting by the fireplace and warming ourselves while something else was happening. That background, continuous feeling is exactly what the imperfective грелись expresses.

У usually means by, near, or at, and it requires the genitive case.

So:

• dictionary form: камин = fireplace
• after у: камина

That is why the sentence has у камина = by the fireplace.

Compare:

• у окна = by the window
• у двери = by the door
• у моря = by the sea

So both после and у are prepositions in this sentence that trigger the genitive.

Here пока means while.

So пока чайник кипел means while the kettle was boiling.

This word is very common, but it can mean different things depending on context:

• Пока! = Bye!
• Пока не... = until
• пока in this sentence = while

So here it introduces a time clause showing that two actions were happening at the same time:

• мы сидели и грелись
• чайник кипел

Because кипел describes an ongoing process: the kettle was boiling.

• кипеть = to boil, to be boiling
• кипел = was boiling

But закипеть means to start boiling / come to a boil.

So:

• чайник кипел = the kettle was boiling
• чайник закипел = the kettle started boiling / came to a boil

In this sentence, кипел fits better because it matches the continuous background action introduced by пока.

Because пока чайник кипел is a subordinate clause.

Russian uses commas to separate the main clause from a subordinate clause much like English often does.

Main clause:

• После долгой прогулки мы сидели у камина и грелись

Subordinate clause:

• пока чайник кипел

So the comma marks the boundary between:

• what we were doing
• what was happening at the same time

No, Russian word order is relatively flexible, though some versions sound more natural than others.

The original sentence is neutral and natural:

• После долгой прогулки мы сидели у камина и грелись, пока чайник кипел.

You could move parts around for emphasis, for example:

• Мы после долгой прогулки сидели у камина и грелись, пока чайник кипел.
• Пока чайник кипел, мы сидели у камина и грелись после долгой прогулки.

These are still understandable, but the original order sounds smoother and more standard.

Russian often uses word order to control focus and emphasis, not just grammar.

Yes, in this context чайник most naturally means kettle.

The verb кипел strongly supports that meaning, because kettles boil. In some contexts чайник can also refer to a teapot, but in everyday modern Russian, especially with кипеть, learners should usually understand it as kettle.

So in this sentence:

• чайник кипел = the kettle was boiling