После долгой прогулки мне хочется полежать на диване.

Because после always requires the genitive case.

• Dictionary form: прогулка
• Genitive singular: прогулки

So:

• после прогулки = after a walk
• после долгой прогулки = after a long walk

This is a very common pattern in Russian:

• после работы = after work
• после урока = after the lesson
• после обеда = after lunch

Because the adjective has to match the noun it describes in gender, number, and case.

Here, прогулки is:

• feminine
• singular
• genitive

So the adjective must also be feminine singular genitive:

• nominative: долгая прогулка
• genitive: долгой прогулки

That is why the sentence has после долгой прогулки.

Because хочется is used in an impersonal construction, where the person experiencing the feeling is put in the dative case.

So:

• мне хочется = I feel like...
• literally, something like to me it is wanted / I have the desire

That is why Russian uses:

• мне = to me
• not я = I

More examples:

• Мне хочется спать. = I feel sleepy / I want to sleep.
• Ему хочется есть. = He feels like eating.
• Нам хочется домой. = We feel like going home.

Хочется means something like:

• I feel like
• I want to
• I have a desire to

But it is usually a bit softer and more feeling-based than я хочу.

Compare:

• Я хочу полежать на диване. = I want to lie on the couch.
  
  • more direct, straightforward
• Мне хочется полежать на диване. = I feel like lying on the couch.
  
  • more natural for a passing desire or mood

So in this sentence, мне хочется suggests a comfortable, personal urge after the walk.

Because полежать is the perfective form, and here it suggests lying down for a while.

Compare:

• лежать = to be lying / to lie (imperfective, general or ongoing)
• полежать = to lie down for some time (perfective, limited duration)

So:

• мне хочется лежать на диване can sound more general, like the activity itself
• мне хочется полежать на диване means I feel like lying on the couch for a bit

This use of по- often gives the sense of doing something for a short time:

• посидеть = sit for a while
• постоять = stand for a while
• погулять = take a walk / walk for a while
• полежать = lie down for a while

Because this sentence describes location, not movement toward a place.

With на:

• на диване = on the couch / on the sofa → location
• на диван = onto the couch → direction or movement

Here the idea is to lie on the couch, so Russian uses the prepositional case:

• диван → на диване

Compare:

• Я сажусь на диван. = I sit down onto the couch.
• Я сижу на диване. = I am sitting on the couch.

Not in the usual English sense. This is an impersonal sentence.

The core of it is:

• хочется полежать = there is a desire to lie down for a while

The person who feels this desire is not the grammatical subject; it appears in the dative:

• мне хочется = I feel like...

So Russian does not need a subject like я here.

It sets the time/background context: after a long walk.

Russian often puts this kind of phrase at the beginning to frame the situation first:

• После долгой прогулки мне хочется полежать на диване.

This word order sounds natural and neutral. It gives the listener the context first, then the feeling, then the action.

You could rearrange the sentence, but the emphasis would change. For example:

• Мне хочется полежать на диване после долгой прогулки.

This is also grammatical, but it places the after a long walk part later and can sound slightly less neutral depending on context.

Yes, you can, but the nuance is a little different.

• Я хочу полежать на диване.
  = I want to lie on the couch.

  • direct
  • clear statement of intention

• Мне хочется полежать на диване.
  = I feel like lying on the couch.

  • softer
  • more emotional or spontaneous
  • often used for a temporary feeling

In this sentence, мне хочется fits very well because it describes how someone feels after a long walk.

Because Russian infinitives do not need a separate word like English to.

So:

• полежать already means to lie down / to lie for a while
• there is no extra particle needed

Other examples:

• Я хочу спать. = I want to sleep.
• Он любит читать. = He likes to read.
• Нам надо идти. = We need to go.

It would still be natural, but the meaning would shift slightly.

• Мне хочется полежать на диване.
  = I feel like lying on the couch for a while.

  • focuses on the action of lying down

• Мне хочется отдохнуть.
  = I feel like resting.

  • broader idea: relaxing, recovering energy

You could also say:

• После долгой прогулки мне хочется отдохнуть.

That means the person wants rest in general, while полежать на диване paints a more specific picture.