Мы хотим опередить их и отыграться.

Yes, хотим already shows 1st person plural, so мы is not strictly necessary.

• Хотим опередить их и отыграться. = perfectly natural
• Мы хотим опередить их и отыграться. = also natural

Adding мы can:

• make the subject more explicit
• add emphasis or contrast, as in we want to do this

Russian often drops subject pronouns when they are clear from the verb form.

After хотеть (to want), Russian normally uses an infinitive to say what someone wants to do.

So:

• хотим опередить = want to get ahead / overtake
• хотим отыграться = want to come back / get even / recover

This is very similar to English want to do something.

They are perfective because the sentence is about achieving results, not just being engaged in the process.

• опередить = to get ahead of / overtake successfully
• отыграться = to recover / come back / win back what was lost

With хотеть, both imperfective and perfective are possible, but they mean slightly different things:

• хотим опережать их = we want to be ahead of them regularly / as an ongoing state
• хотим опередить их = we want to get ahead of them, to achieve that result

And:

• хотим отыгрываться would sound like wanting to be in the process repeatedly
• хотим отыграться means wanting to make a comeback / recover successfully

So the perfective fits the idea of a definite goal.

Because их is the object form, while они is the subject form.

• они = they
• их = them

In this sentence, их is the direct object of опередить:

• опередить их = to overtake them

So они would be wrong here.

Because опередить takes a direct object in the accusative case.

For plural animate nouns and pronouns, the accusative form looks like the genitive form. That is why we get:

• они → их

This is a very common Russian pattern:

• я вижу их = I see them
• мы знаем их = we know them
• опередить их = overtake them

So although learners often memorize их as their / them, here it is specifically accusative plural meaning them.

Not exactly.

Опередить means something like:

• get ahead of
• overtake
• outpace
• move in front of

It focuses on being ahead of someone, often in a race, competition, ranking, or metaphorical contest.

Победить means to defeat / to win against someone.

So:

• опередить их = get ahead of them
• победить их = defeat them

You can overtake someone without fully defeating them, depending on context. That is why опередить is more specific.

Отыграться is a tricky but common verb. It often means:

• to recover after being behind
• to make a comeback
• to win back what was lost
• sometimes, more loosely, to get even

In sports or games, it often means coming back after losing or trailing.

So in this sentence, it suggests not just doing well, but reversing a bad situation.

That is the key nuance: отыграться usually implies that things were going badly before.

The ending -ся marks the verb as reflexive, but reflexive verbs in Russian are not always literally reflexive in the English sense.

Here отыграться is simply the dictionary form of this verb with its own meaning. You should learn it as a whole word.

Compare:

• отыграть = to play back / perform / win back something
• отыграться = to recover, make a comeback, get even

So the -ся is not something you can ignore. It changes the meaning of the verb.

Because отыграться can be used on its own when the context already makes clear what is meant.

Here it means something like:

• make a comeback
• recover
• get even

Russian often leaves such things implicit when they are understood from the situation.

Also, this verb can appear in other patterns:

• отыграться после поражения = to bounce back after a defeat
• отыграться на ком-то = to take it out on someone

That last pattern is different and has a different nuance. In your sentence, отыграться is being used more generally, without naming a target.

Because и is simply joining two infinitives that belong to the same verb хотим:

• хотим опередить
• (хотим) отыграться

This is a normal coordinated structure, so no comma is needed:

• Мы хотим опередить их и отыграться.

A comma would usually appear only if there were separate clauses or some other reason for punctuation.

Yes, very naturally.

The verb хотим applies to both infinitives:

• хотим опередить
• хотим отыграться

Russian often avoids repeating the same verb when it is understood. English does the same:

• We want to overtake them and make a comeback.

So this structure is completely standard.

Yes. Russian word order is flexible, although the basic version here is very natural.

Possible variants include:

• Мы хотим опередить их и отыграться.
• Хотим опередить их и отыграться.
• Их мы хотим опередить и отыграться. — stronger emphasis on them
• Отыграться и опередить их мы хотим. — marked, more stylistic or contrastive

The original sentence is the most neutral and straightforward.

A learner may want to know the imperfective partners:

• опережать → опередить
• отыгрываться → отыграться

Very roughly:

• imperfective = process, repetition, general action
• perfective = completed result, single achievement

So:

• опережать = be ahead of / keep overtaking
• опередить = get ahead of
• отыгрываться = be coming back / keep recovering
• отыграться = recover successfully / make a comeback

In your sentence, the perfective forms fit the idea of reaching a goal.

A natural pronunciation would be:

Мы хоти́м опереди́ть их и отыгра́ться.

Main stress:

• хоти́м
• опереди́ть
• отыгра́ться

A few pronunciation notes:

• их is usually pronounced like ikh
• in fast speech, мы can sound quite light
• the stress in отыгра́ться is on -гра́-

If you say it smoothly, the sentence has a natural rhythm: Мы хоти́м опереди́ть их и отыгра́ться.