Como utilizar do, does, doesn't, isn't, aren't, not

"Do" é usado com os pronomes I, you, we, they. É usado em perguntas que se refere a uma ação fixa praticada pela pessoa.

Ex: Do you like apples? (Você gosta de maçã?)

"Does" é usado da mesma forma, porém com os pronomes he, she, it.

Ex: Does he like apples? (Ele gosta de maçã?)

Qualquer pergunta que involve algum verbo você vai usar esses auxiliares. Você faz...?/você fala...?/você joga...?, etc.

Quando se utiliza o "are" ou "is" em perguntas significa que se refere a um contexto que pergunta se a pessoa "é" ou "está". Ao contrário dos auxiliares DO e DOESN'T que são sobre "fazer". O "are" sendo usado pra you, we, they, e "is" pra he, she, it.

Ex: Are you a football player? (Você é jogador de futebol?)
Is he a football player? (Ele é jogador de futebol?)

As frases na negativa de afirmação
com essa estrutura segue o mesmo contexto, porém você tá afirmando e negando.

Ex: You aren't a football player/he isn't a football player (você não é jogador de futebol/ele não é...)

O "not" é apenas outra forma de usar o isn't. He isn't = he's not.

Leonardo;
Por qual razão na frase abaixo usa-se o not e não o doesn't?
In the street below could catch the ball and not give it back.

Obrigado pela atenção.

NOT there is used as a conjunction, a coordinator word.
It is a coordinator conjunction since it coordinates the two sentences "in the street below could catch the ball" to not "give it back."
Of course, it works along with AND in the case (there is, it is cumulative 'AND NOT') in the given sentence.

I am no grammarian, so I could be mistaken. Let's wait and see if advanced learners, teachers, or others agree or have other insights to add to the topic.

Ref. en.wikipedia

German Alexandre escreveu: 14 Mai 2022, 14:34 Leonardo;
Por qual razão na frase abaixo usa-se o not e não o doesn't?
In the street below could catch the ball and not give it back.

Obrigado pela atenção.

We can also use "NOT" in the following situations: to express contrast; to negate a to-infinitive clause; or to negate a non-finite clause.

1) Expressing contrast
One of the functions of "NOT" is to express contrast, and you can omit the verb phrase or part of it in the second clause. It usually comes with "but" as in "but not" :

- You can look but can't touch it. (the pronoun "you" was left out in the second clause)
- Drive the yellow car, but not the blue one. (the verb "drive" was left out)
- There were a few problems, but not too many.

2) Negating a to-infinitive clause
-She tried to give it back. ("to give it back " is a to-infinitive clause.)
-She tried not to give it back. (this is how you negate it.)

3) Negating a non-finite clause
Non-finite clauses are clauses without subject where the main verb is in the following forms: to-infinitive, -ing form or -ed form. To make a negative statement, we use "NOT".

to-infinitive clause
- To make it harder for me, she decided to quit the contest.
- Not to make it harder for me, she decided to quit the contest.

-ing form clause
- Drinking a beer in the evening was what I had in mind before I met her.
- Not drinking a beer in the evening was what I had in mind before I met her.

-ed form clause
- Raised in high-society circles, Mary grew accustomed to it.
- Not raised in high-society circles, Mary grew accustomed to it.

Well, as of your exemple, I believe it is not according to the proper grammar rules of the English language, once it lacks the subject of the sentence and the second clause is missing the "to" in order to form the to-infinite clause "to give it back". In a formal context, it should be written like this:

- In the street below, I could catch the ball and not to give it back.