A equipe vai resgatar a gente.

They are doing different jobs:

• In a equipe, a is the definite article, meaning the.
• In a gente, a is part of the fixed expression a gente.

So they are not the same grammatical element, even though they look identical.

Because the subject is a equipe, which is grammatically singular.

Even though a team is made up of several people, Portuguese normally treats equipe as a singular noun, so the verb must also be singular:

• A equipe vai... = correct
• As equipes vão... = plural subject, so plural verb

This is the very common Brazilian Portuguese future pattern:

• ir + infinitive
• vai resgatar = literally is going to rescue, but in practice often just will rescue

In everyday Brazilian Portuguese, this structure is usually more common than the simple future form.

Yes. That is grammatically correct.

But there is a style difference:

• A equipe vai resgatar a gente = more common in everyday speech
• A equipe resgatará a gente = more formal, more written, less conversational

So learners will hear vai + infinitive much more often in normal spoken Brazilian Portuguese.

Resgatar often means to rescue, to save, or to recover/retrieve, depending on context.

Common uses include:

• rescuing people from danger
• recovering something lost or taken
• redeeming something in some contexts

In this sentence, it has the clear sense of rescue.

Yes, a gente is very often used in Brazilian Portuguese to mean we in informal speech:

• A gente vai = We are going

But in colloquial Brazilian Portuguese, a gente can also refer to us after a verb:

• A equipe vai resgatar a gente = The team will rescue us

So here, a gente is the object of resgatar.

This is very natural in spoken Brazilian Portuguese.

Yes. You could say:

• A equipe vai nos resgatar

That also means the same thing.

The difference is mostly one of style and usage:

• a gente = very common and natural in everyday Brazilian Portuguese
• nos = correct, but often feels more formal, more written, or less conversational in many contexts

Also, nós would not be the right choice here, because nós is a subject form, not the normal direct object form.

Because equipe is the more general word for team or crew, especially outside sports.

• equipe = team, staff, crew, group working together
• time = usually a sports team

So for a rescue team, emergency team, medical team, or work team, equipe is the natural choice.

The normal order here is:

• subject + vai + infinitive + object
• A equipe vai resgatar a gente

With a gente, this is the natural placement.

If you use nos, you can say:

• A equipe vai nos resgatar

But something like A equipe vai a gente resgatar is not natural standard word order.

A simple English-friendly approximation is:

• ah eh-KEE-pee vai hez-gah-TAR ah ZHEN-chee

A few helpful pronunciation notes:

• equipe sounds roughly like eh-KEE-pee
• vai sounds a bit like vye
• the r at the start of resgatar sounds like an English h in most of Brazil
• gente begins with the zh sound, like the s in measure
• the first syllable of gente is nasal, so it is not a plain English gen

A more natural connected rhythm would sound something like:

• a equipe vai resgatar a gente
• ah-eh-KEE-pee vai hez-gah-TAR-ah ZHEN-chee