Uncle said: “He has to go”.

Uncle said he had to go.

Aunt replied, “Let him stay”.

Aunt asked him to stay.

Sister asked, “Why does he have to go?”.

Привет, Роман! В предложении Aunt replied, “Let him stay” тот, кого и о ком она просит - два разных человека. Т.е она попросила не кого-то остаться, а чтобы кому-то разрешили остаться.

Sister asked why he had to go?”.

She yelled, “Don't bother me!”.

Здравствуй, Алексей.
Aunt replied, “Let him stay”. - Тетя ответила: «Пусть он останется»
Aunt asked him to stay. - Тетя попросила (что бы ему позволили) остаться. ⇒ Тетя попросила ему остаться.
Если бы в исходной фразе было больше информации о том, кого она просит, то можно было бы более подробно ответить.

She yelled (to him) not to bother her.

Grandfather said softly, “Don't look at me like that”.

Можно сделать обезличенным пассивом в present subjunctive: She asked he (should) be allowed to stay. Либо с for + инфинитив: She asked for him to be allowed to stay.

Grandfather said softly he should not be looked at like that.

She begged him, “Tell me the truth now”.

She begged him to tell her the truth now.

She asked, “Do you speak English?”

She asked him/her if he/she spoke English.

“I didn't see him yesterday”, Bill retorted.

Bill retorted that he hadn't seen him the day earlier.

She said to them, “Don't stay there!”.

She told them not to stay there.

“And yet it moves”, said Galilei.

Galilei said that it moves yet.

The teacher said, “The lesson is over”.

The teacher said that the lesson was over.

He asked me, “Are you all right?”.

He asked me if I was all right.

The boss asked Sam, “Whose work has she done?”

The boss asked Sam whose work she had done.

Bill said to John, “If I were you, I wouldn't have bought a car”.