Ребёнка тошнило всю ночь.

Because this is an impersonal construction.

With тошнить meaning to feel sick / to be nauseous / to be vomiting, Russian usually does not make the person the grammatical subject. Instead, the person experiencing the condition is put in the accusative:

• Меня тошнит.
• Его тошнило.
• Ребёнка тошнило.

So ребёнка here is not the subject. It is the person affected by the action/state.

Here it is accusative.

The confusion happens because animate masculine singular nouns often have an accusative form that looks exactly like the genitive.

For ребёнок:

• nominative: ребёнок
• genitive: ребёнка
• accusative (animate): ребёнка

So the form is the same as the genitive, but the function in this sentence is accusative.

Because the sentence is impersonal.

In Russian, impersonal past-tense predicates are commonly put in the neuter singular, even when a real person is involved in the meaning.

So:

• Меня тошнило.
• Её знобило.
• Ребёнка трясло.

The -ло ending does not mean the child is grammatically neuter here. It just shows the normal past-tense form used in this kind of subjectless sentence.

Because Russian uses the accusative to express duration of time.

So:

• весь день = all day
• всю ночь = all night
• всю неделю = all week

In this sentence, всю ночь means for the whole night / throughout the night.

Normally, no.

In standard Russian, тошнить in this bodily-state meaning is usually used impersonally:

• natural: Ребёнка тошнило всю ночь
• not the usual standard pattern: Ребёнок тошнил всю ночь

If you want a sentence with ребёнок as the grammatical subject, Russian usually prefers another verb or expression, depending on the exact meaning:

• Ребёнка рвало всю ночь = the child was vomiting all night
• Ребёнок плохо себя чувствовал всю ночь = the child felt unwell all night

So with тошнить, the impersonal pattern is the normal one.

They are close in meaning, but not identical.

• тошнило = there was an ongoing period of nausea; in context it can also imply being sick repeatedly
• стошнило = a completed event, usually threw up once
• рвало = strongly suggests actual vomiting, often repeatedly

Compare:

• Ребёнка тошнило всю ночь = the child felt sick / was being sick all night
• Ребёнка стошнило = the child threw up
• Ребёнка рвало всю ночь = the child was vomiting all night

The construction stays the same.

• present: Ребёнка тошнит.
• past: Ребёнка тошнило.
• future: Ребёнка будет тошнить.

The person affected still stays in the accusative:

• Меня тошнит.
• Меня тошнило.
• Меня будет тошнить.

No. Russian word order is fairly flexible.

Ребёнка тошнило всю ночь is a natural, neutral order. But you can move things around for emphasis:

• Всю ночь ребёнка тошнило = emphasizes all night
• Ребёнка всю ночь тошнило = also natural

The core meaning does not change much, but the focus can shift depending on what comes first.