Я так и не привык к тому, что у собаки всегда мокрый нос и холодные лапы после прогулки.

Так и не is a very common Russian pattern meaning something like:

• never did
• still didn’t
• ended up not

So Я так и не привык means more than just I didn’t get used to it. It suggests that the speaker expected that maybe they would get used to it, but in the end, that never happened.

Compare:

• Я не привык = I didn’t get used to it / I’m not used to it
• Я так и не привык = I never did get used to it

It often adds a feeling of mild frustration, surprise, or resignation.

Here привык is the past tense masculine singular form of the perfective verb привыкнуть.

So:

• привыкнуть = to get used to, to become accustomed to
• привык = got used to / became accustomed to

But in this sentence, because of так и не, the meaning is:

• так и не привык = never got used to

Why not привыкал?

• привыкал is imperfective and means was getting used to / used to get used to
• It focuses on the process, not the result

Here the speaker is talking about the final outcome: it never happened. That is why the perfective form fits better.

Why not привыкнуть?

• привыкнуть is the infinitive, so it cannot be the main finite verb here
• The sentence needs a past-tense form, so привык is used

The verb привыкнуть / привыкать requires к + dative:

• привыкнуть к чему? = to get used to what?

When what follows is not just a noun, but a whole clause, Russian often uses:

• к тому, что... = to the fact that...

So:

• Я привык к шуму = I got used to the noise
• Я привык к тому, что здесь шумно = I got used to the fact that it is noisy here

In your sentence:

• Я так и не привык к тому, что...
• literally: I never got used to the fact that...

This is a very standard construction.

Because of the preposition к, which requires the dative case.

The pronoun то changes by case:

• nominative: то
• genitive: того
• dative: тому
• etc.

Since привыкнуть к takes dative, you get:

• к тому

So к тому, что... is grammatically required here.

Russian very often expresses possession with the structure:

• у + genitive + noun
• literally: at someone there is...
• naturally translated as someone has...

So:

• у собаки мокрый нос = the dog has a wet nose
• literally: by the dog [there is] a wet nose

This is much more natural in Russian than using иметь in everyday speech.

Why собаки?

Because у requires the genitive case:

• собака → nominative
• собаки → genitive singular

So у собаки means the dog has.

In the possession structure у кого-то есть / был / будет / Ø что-то, the thing possessed is normally in the nominative case.

So in:

• у собаки мокрый нос и холодные лапы

the possessed things are:

• нос
• лапы

Both are nominative.

That is why the adjectives also appear in nominative:

• мокрый нос
• холодные лапы

This is different from English, where possession is shown with have. In Russian, this structure behaves more like there is/there are.

In the present tense, Russian often omits есть when it just means is/are in simple existence or possession statements.

So:

• У собаки есть мокрый нос is possible, but slightly different
• У собаки мокрый нос is very natural and common

In many present-tense sentences, Russian simply leaves the linking verb out:

• Он врач = He is a doctor
• У меня машина = I have a car
• У собаки мокрый нос = The dog has a wet nose / The dog’s nose is wet

Here zero copula is normal.

Grammatically, it is a possession/existence construction.

A rough literal breakdown:

• у собаки = by the dog / the dog has
• всегда = always
• мокрый нос = a wet nose
• и холодные лапы = and cold paws

So the clause means:

• that the dog always has a wet nose and cold paws

Russian often uses this structure instead of directly saying the dog’s nose is wet and its paws are cold.

Because the adjectives must agree with the nouns in gender, number, and case.

• нос is masculine singular, nominative
  → мокрый нос
• лапы is plural, nominative
  → холодные лапы

So the adjective endings are different because the nouns are different.

Yes, normally it applies to the whole statement:

• the dog always has a wet nose
• and always has cold paws after a walk

So всегда is understood with both parts.

Russian often places an adverb once, and it naturally covers the coordinated items that follow.

Here после прогулки most naturally modifies the inner clause:

• у собаки всегда мокрый нос и холодные лапы после прогулки

That means:

• after a walk, the dog has a wet nose and cold paws

It does not normally mean:

• I got used to it after walks

So the idea is that the dog’s nose and paws are like that after being walked.

In English, you might naturally place the phrase earlier to make this clearer:

• ...that after a walk the dog always has a wet nose and cold paws.

Because the preposition после requires the genitive case.

So:

• прогулка = nominative
• прогулки = genitive singular

Therefore:

• после прогулки = after a walk / after the walk

This is just standard case government after после.

Russian does not have articles, so после прогулки can mean either:

• after a walk
• after the walk

The exact interpretation depends on context.

In this sentence, English would probably most naturally say:

• after a walk

because it sounds like a general habit or repeated situation, not one specific walk.

Yes, but к тому, что is the most natural way when the object of привыкнуть is a whole clause.

For example, you might also see something more compact with a noun phrase:

• Я так и не привык к мокрому носу и холодным лапам собаки после прогулки.

That is possible, but it changes the style and structure. It turns the idea into a noun phrase rather than the fact that...

The original sentence sounds very natural because it presents the whole situation as something the speaker could not get used to.

Yes, Russian word order is flexible.

The given sentence is natural:

• Я так и не привык к тому, что у собаки всегда мокрый нос и холодные лапы после прогулки.

But you could move things around for emphasis. For example:

• Я так и не привык к тому, что после прогулки у собаки всегда мокрый нос и холодные лапы.

This version makes после прогулки attach more clearly to the dog’s condition.

You could also emphasize the dog:

• ...что у собаки после прогулки всегда мокрый нос и холодные лапы.

So the word order can change, but the original is still understandable and natural.

It strongly implies that the speaker is still not used to it now.

Although привык is a past tense form, the expression так и не привык usually means:

• the expected change never happened
• and the situation remains unresolved

So in practice, it often corresponds to English:

• I still never got used to it
• I just never got used to it

It is about a past process whose result never arrived, and that has consequences in the present.

Not exactly.

• Я так и не привык = I never got used to it
• Я не могу привыкнуть = I can’t get used to it

The second one focuses more on the current struggle or inability. The first one looks back and says that, despite time or expectation, it never happened.

Both can be natural, but they are not identical.

Because собака is grammatically a feminine noun, regardless of the actual sex of the animal in many contexts.

So:

• у собаки
• собака была
• etc.

Russian grammatical gender does not always match biological sex. If the speaker wanted to emphasize a male dog specifically, they might use пёс, but собака is a general everyday word for dog.