Я хочу утешить друга после неудачного выступления.

Because it follows хочу = I want. After хочу, Russian normally uses an infinitive to say what someone wants to do:

• Я хочу утешить друга. = I want to comfort my friend.
• Я хочу помочь. = I want to help.
• Я хочу уйти. = I want to leave.

So утешить is the basic to comfort form that depends on хочу.

This is an aspect question.

• утешить is perfective
• утешать is imperfective

Here, утешить sounds natural because the speaker wants to do a single, complete action: to comfort the friend after the bad performance.

So:

• Я хочу утешить друга = I want to comfort my friend
  (one concrete act, with a result in mind)

If you said:

• Я хочу утешать друга

it would sound more like I want to be comforting my friend / I want to comfort him repeatedly or over a period of time, which is less likely in this context.

Because друга is the accusative singular form of an animate masculine noun.

In Russian, animate masculine nouns often have:

• nominative: друг = friend
• accusative: друга = friend as a direct object

Since the friend is the person being comforted, друг is the direct object, so Russian uses the accusative:

• Я вижу друга. = I see my friend.
• Я хочу утешить друга. = I want to comfort my friend.

This is a very common pattern with masculine animate nouns.

Because после always requires the genitive case.

So after после = after, the following noun phrase must go into the genitive:

• после урока = after the lesson
• после концерта = after the concert
• после выступления = after the performance/speech

That is why we get:

• выступление → выступления
• неудачное выступление → неудачного выступления

неудачного is the adjective неудачный in the genitive singular, agreeing with выступления.

Here is the base form:

• неудачное выступление = an unsuccessful performance

But after после, the phrase has to be genitive:

• после неудачного выступления

The adjective must match the noun in case, number, and gender:

• noun: выступления = genitive singular
• adjective: неудачного = genitive singular form matching it

So the ending is there because of agreement.

Выступление is a broad word. Depending on context, it can mean:

• a speech
• a presentation
• a performance
• a public appearance

In this sentence, it probably means some kind of public performance or presentation that did not go well.

So после неудачного выступления could mean:

• after an unsuccessful speech
• after a bad presentation
• after a poor performance

The exact English translation depends on context.

Yes. Russian often drops the subject pronoun when it is clear from the verb ending.

• Я хочу утешить друга...
• Хочу утешить друга...

Both are correct. The full version with Я may sound a little more explicit or emphatic, while the version without Я is often more natural in everyday speech.

No, Russian word order is fairly flexible, although some orders sound more neutral than others.

The most neutral version is:

• Я хочу утешить друга после неудачного выступления.

But you could also say:

• После неудачного выступления я хочу утешить друга.
• Я после неудачного выступления хочу утешить друга.

These all mean roughly the same thing, but the emphasis changes:

• putting После неудачного выступления first highlights the time/context
• the original order is the most straightforward and neutral

Russian often leaves out possessive words like my when they are obvious from context.

So:

• Я хочу утешить друга literally looks like I want to comfort friend
• but in natural English it is often understood as I want to comfort my friend

If you want to be explicit, you can say:

• Я хочу утешить моего друга.

That is also correct, but it can sound more specific or emphatic. In many everyday situations, just друга is enough.

Yes, and that is very common.

• Я хочу утешить своего друга = I want to comfort my friend

Russian often uses свой to mean one’s own when the possessor is the subject of the sentence. Since я is the subject and the friend belongs to me, своего друга works very naturally.

So you may hear:

• Я хочу утешить друга
• Я хочу утешить своего друга
• Я хочу утешить моего друга

All are possible, but своего друга is especially natural when the friend is the subject’s own friend.

Неудачный means something like:

• unsuccessful
• bad
• unfortunate
• that did not go well

So неудачное выступление is a performance, speech, or presentation that went badly.

It does not necessarily mean total disaster; it just means the event was not successful.

A helpful pronunciation guide is:

Ya kha-CHOO oo-TYE-shit' DROO-ga POS-lye nee-oo-DACH-na-va vys-too-PLYE-ni-ya.

Approximate stress:

• Я хочу́ уте́шить дру́га по́сле неуда́чного выступле́ния.

A few useful notes:

• хочу́ has stress on the last syllable
• уте́шить has stress on те
• дру́га has stress on дру
• по́сле has stress on the first syllable
• неуда́чного has stress on да
• выступле́ния has stress on ле

Yes, but the meaning would change a little.

• утешить = to comfort, console
• успокоить = to calm down, reassure
• поддержать = to support

So:

• Я хочу утешить друга... = I want to comfort my friend emotionally
• Я хочу успокоить друга... = I want to calm my friend down
• Я хочу поддержать друга... = I want to support my friend

In the context of a bad performance, утешить is a very natural choice if the friend feels upset or disappointed.