How Children Lived at the Time of Prophet Muhammad
Question

How were children brought up at the time of the Prophet? What were some boy-specific and girl-specific games/activities? Which chores did each child help their parents with (and at what age did they start to help out)? Did all girls remain at home? If not, what jobs did they do outside (like trading, etc.)? Can you describe a typical day in a child’s life?
Praise be to Allah.A typical day in a child’s life at the time of Prophet Muhammad
For children who were seven years old at the time of the Prophet (blessings and peace of Allah be upon him), it seems that their day would begin with Fajr prayer , because the Prophet (blessings and peace of Allah be upon him) used to enjoin his companions to start teaching their children how to pray when they reached the age of seven years.
It was narrated from `Amr ibn Shu`ayb, from his father, that his grandfather said: The Messenger of Allah (blessings and peace of Allah be upon him) said: Instruct your children to pray when they are seven years old, and smack them if they do not pray when they are ten years old, and separate them in their beds.” (Narrated by Abu Dawud, 459. Also narrated by Abu Dawud, 494 and At-Tirmidhi, 407 from Sabrah ibn Ma`bad. At-Tirmidhi said: The Hadith of Sabrah ibn Ma`bad Al-Juhani is a sound Hadith)
The Companions (May Allah be pleased with him) used to fill their children’s day with three things:

They would teach them faith and Islam, according to whatever each parent was able to do.

It was narrated that Jundub ibn `Abdullah (may Allah be pleased with him) said: We were with the Prophet (blessings and peace of Allah be upon him), and we were strong boys, so we learnt faith before we learnt the Quran. Then we learnt the Quran and our faith increased thereby. (Narrated by Ibn Maajah, 61; classed as authentic by Al-Albani in Sahih Sunan Ibn Majah, 1/37-38)
Ibn Al-Athir (may Allah have mercy on him) said:
[The word translated here as strong] may also refer to a boy who is approaching puberty.”(An-Nihayah fi Gharib Al-Hadith, 1/380)
It was narrated from Ibn `Abbas (may Allah be pleased with him): When the Messenger of Allah (blessings and peace of Allah be upon him) died, I was ten years old, and I had read Al-Muhkam. (Narrated by Al-Bukhari, 5035)
“Al-Muhkam” refers to the surahs of Al-Mufassal (from Qaf or Al-Hujurat – according to different scholarly views – to Surat an-Nas)
It was narrated that Al-Bara’ ibn `Azib (may Allah be pleased with him) said: The first ones to come to us were Mus`ab ibn `Umayr and Ibn Umm Maktum, who taught the people the Quran. Then came Bilal, Sa`d and `Ammar ibn Yasir. Then `Umar ibn Al-Khattab came with twenty of the companions of the Prophet (blessings and peace of Allah be upon him). Then the Prophet (blessings and peace of Allah be upon him) came, and I never saw the people of Madinah rejoice at anything as much as they rejoiced at the arrival of the Messenger of Allah (blessings and peace of Allah be upon him). Even the slave girls were saying: The Messenger of Allah has come! Before he arrived, I had read {Sabbih isma rabbik Al-a`la (Glorify the name of your Lord, the Most High)} [Surat Al-A`la], along with other surahs of Al-Mufassal. (Narrated by Al-Bukhari, 3925)
At that time, Al-Bara’ (may Allah be pleased with him) was still a child, because he was deemed too young to participate in the battle of Badr.
When a child reached the age of seven, his parents would watch him to make sure that he did all of the five daily prayers, as is mentioned in the Hadith quoted above. And perhaps some children also offered voluntary prayers, according to what each of them was able to do.
It was narrated that Ibn `Abbas (may Allah be pleased with him) said: I stayed overnight with my maternal aunt, and the Prophet (blessings and peace of Allah be upon him) got up to pray at night, so I got up to pray with him. I stood on his left, and he took hold of my head and made me stand on his right. (Narrated by Al-Bukhari, 699)
Perhaps some of them would fast, in order to get used to it so that it would be easy for them when they grew up.
It was narrated that Ar-Rubayyi` bint Mu`awwidh (may Allah be pleased with her) said: On the morning of `Ashura, the Prophet (blessings and peace of Allah be upon him) sent word to the villages of the Ansar around Madinah, saying: “Whoever started the day not fasting, let him complete the rest of the day (without food), and whoever started the day fasting , let him complete his fast.”
After that, we used to fast on this day, and we would make our children fast too . We would make them toys out of wool, and if one of them cried for food, we would give (that toy) to him until it was time to break the fast. (Narrated by Al-Bukhari, 1960 and Muslim, 1136)
Perhaps some of them also did Hajj.
It was narrated that As-Sa’ib ibn Yazid (may Allah be pleased with him) said: I was taken to do Hajj with the Messenger of Allah (blessings and peace of Allah be upon him) when I was seven years old. (Narrated by Al-Bukhari, 1858)

The children would help out their families with whatever they were able to do of day-to-day tasks and service.

Anas (may Allah be pleased with him) said: When the Messenger of Allah (blessings and peace of Allah be upon him) came to Madinah, he had no servant. Abu Talhah took me by the hand and brought me to the Messenger of Allah (blessings and peace of Allah be upon him), and said: O Messenger of Allah, Anas is a good boy, let him serve you. I served him whilst travelling and at home, and by Allah he never said to me about something that I had done: Why did you do this like this? Or for something that I had not done: Why did you not do this like this? (Narrated by Al-Bukhari, 2768) and Muslim, 2309)
His age when he started to serve the Prophet (blessings and peace of Allah be upon him) was ten years.
It was narrated from Anas ibn Maalik (may Allah be pleased with him) that he was ten years old when the Messenger of Allah (blessings and peace of Allah be upon him) came to Madinah. [He said:] My mother and aunts used to regularly urge me to serve the Prophet (blessings and peace of Allah be upon him), and I served him for ten years. The Prophet (blessings and peace of Allah be upon him) died when I was twenty years old. (Narrated by Al-Bukhari, 5166)

They also had their fair share of play and fun.

It was narrated that `A’ishah (may Allah be pleased with her) said: I used to play with dolls in the house of the Messenger of Allah (blessings and peace of Allah be upon him). I had friends who would play with me, but when the Messenger of Allah (blessings and peace of Allah be upon him) came in, they would hide themselves from him, but the Messenger of Allah (blessings and peace of Allah be upon him) would call them to play with me. (Narrated by Al-Bukhari, 6130 and Muslim, 2440)
Anas (may Allah be pleased with him) said: The Messenger of Allah (blessings and peace of Allah be upon him) was one of the best people in attitude. One day he sent me to do an errand for him, and I said: By Allah, I will not go. But in my heart I intended to go and do what the Prophet of Allah (blessings and peace of Allah be upon him) had told me to do. Then I went out and passed by some boys who were playing in the marketplace. Then the Messenger of Allah (blessings and peace of Allah be upon him) caught me on the back of my neck from behind. I looked at him and he was smiling. He said: “O Unays, did you go where I told you to go?” I said: Yes, I am going, O Messenger of Allah.
We have not come across any authentic Hadith which describe in detail what kind of games they used to play, but what appears to be the case is that they carried on with the games that they had known during the Jahiliyyah that were not prohibited according to Islamic teachings, and the games of strength that were known at the time, such as wrestling with one another, as is indicated by some Hadiths.
Dr. Jawad `Ali  has spoken in detail about the children’s games that were known to the Arabs at that time, in his book Al-Mufassal fi Tarikh Al-`Arab Qabl Al-Islam, Dar As-Saqi edn., (9/124-126)
Did girls remain at home at the time of Prophet Muhammad?
The basic principle regarding the women of the Companions is that they stayed in their houses, in response to the words of Allah, may He be exalted (interpretation of the meaning):
{And abide in your houses and do not display yourselves as [was] the display of the former times of ignorance.} [Al-Ahzab 33:33]
They did not go out except for necessary errands, or for prayer if one of them wanted to attend it, but they did not crowd with men in the streets and marketplaces.
Ibn Kathir (may Allah have mercy on him) said:
“The words {And abide in your houses} mean: Stay in your houses and do not go out unnecessarily.
Among the needs that are recognised in Islamic teachings is praying in the mosque, subject to certain conditions, as the Messenger of Allah (blessings and peace of Allah be upon him) said: “Do not prevent the female slaves of Allah from attending the mosques of Allah, but let them go out looking scruffy.” According to one report: “and their houses are better for them.” (Tafsir Ibn Kathir,  6/409)
It seems, from what is mentioned in the Hadiths quoted above about training children to get used to Islamic practices from an early age, that the Companions (may Allah be pleased with them) used to prepare their daughters to adhere to Islamic rulings from a young age, so they taught them to observe the etiquette that would instil shyness and chastity in them, in obedience to the command of Allah (interpretation of the meaning):
{O you who have believed, protect yourselves and your families from a Fire whose fuel is people and stones, over which are [appointed] angels, harsh and severe; they do not disobey Allah in what He commands them but do what they are commanded.} [At-Tahrim 66:6]
And Allah knows best.