Мне не хочется выходить из дома, но я всё-таки пойду в библиотеку.

Because Russian uses an impersonal construction here: мне не хочется... literally works like to me, it is not felt as desirable...

So мне is in the dative case and marks the experiencer — the person who feels this lack of desire. The grammatical subject is not я.

This pattern is very common in Russian:

• Мне хочется спать. = I feel like sleeping.
• Мне не хочется есть. = I don’t feel like eating.

A very literal gloss would be: To me, it doesn’t want itself to go out of the house, but the natural English meaning is simply I don’t feel like leaving the house.

Both can mean something like I don’t want to go out, but the tone is different.

• Мне не хочется выходить = I don’t feel like going out

  • softer
  • more about mood, energy, or inclination
  • often temporary

• Я не хочу выходить = I don’t want to go out

  • more direct
  • more about will or decision
  • can sound firmer

In this sentence, мне не хочется is a good choice because it suggests reluctance or lack of motivation, not a strong refusal.

Because мне is not the subject. It is a dative experiencer.

The verb хочется comes from хотеться, which is often used impersonally in the 3rd person singular:

• Мне хочется
• Ему хочется
• Нам хочется

So the verb does not change to match мне. This is normal for impersonal constructions in Russian.

You can think of it as similar to English expressions like it seems to me or it feels to me, where the person is not the grammatical subject of the main verb.

Выходить is imperfective, and выйти is perfective.

After expressions like хочется / не хочется, Russian often prefers the imperfective infinitive when talking about an action in a general or process-like way:

• Мне не хочется выходить из дома. = I don’t feel like going out / leaving the house.

Using выйти is possible, but it would sound more like a single completed act:

• Мне не хочется выйти из дома. = I don’t feel like stepping out of the house.

So выходить here is more natural because the speaker is talking about the general act of going out, not emphasizing one single completed exit.

Из дома means out of the house / from home.

That is different from домой, which means homeward / to home.

Compare:

• выходить из дома = to go out of the house / leave home
• идти домой = to go home

So in this sentence, the idea is leaving the place where the speaker currently is, which is why Russian uses из + genitive:

• из дома

Also note that дома here is the genitive singular of дом, not the adverb дома meaning at home. The form looks the same, but the grammar makes the meaning clear.

Because the speaker is talking about a future decision: I’ll go to the library anyway.

• пойду is the future form of the perfective verb пойти
• it often means I will go / I will set off

Why not the others?

• иду usually means I am going right now, or sometimes a very near future
• буду идти means I will be walking, which focuses on the process, not the simple fact of going

So я всё-таки пойду в библиотеку is the natural way to say I’ll go to the library anyway.

Yes, basically.

Russian distinguishes between motion on foot and motion by vehicle:

• идти / пойти = go on foot
• ехать / поехать = go by transport

So пойду в библиотеку literally suggests going on foot.

In everyday speech, that is very natural if the place is nearby or if transport is simply not the point. If the speaker meant going by bus, car, metro, etc., Russian would more naturally use поеду:

• Я всё-таки поеду в библиотеку.

Because Russian uses different cases after в depending on meaning:

• в + accusative = motion toward / into a place
• в + prepositional = location inside a place

So:

• в библиотеку = to the library
• в библиотеке = in the library

Since the speaker is going there, the accusative is required:

• пойду в библиотеку

Всё-таки means something like:

• all the same
• nevertheless
• still
• anyway
• despite that

It shows a contrast between the two parts of the sentence:

• Мне не хочется выходить из дома = I don’t feel like leaving the house
• но я всё-таки пойду в библиотеку = but I’ll go to the library anyway

So it adds the idea of doing something despite reluctance.

Also, you will often see it written as все-таки in ordinary writing, even though всё-таки is the more precise spelling.

The word order here is quite natural and neutral.

• Мне не хочется выходить из дома sounds like a normal way to present the speaker’s feeling.
• Но я всё-таки пойду в библиотеку also sounds neutral and conversational.

Russian word order is flexible, so you could move things around for emphasis:

• Но всё-таки я пойду в библиотеку.
  Slightly more emphasis on nevertheless.

• Но в библиотеку я всё-таки пойду.
  Emphasizes the library as the destination.

But the original version is probably the most straightforward and natural for a learner to treat as the default.

Because но means but and joins two separate clauses:

• Мне не хочется выходить из дома
• но я всё-таки пойду в библиотеку

In Russian, a comma is normally required before но when it connects two clauses like this.

So the punctuation here is standard and correct.