Exactly as the title says, from what age did a child of a samurai start to join his father in battles? I suspect it is when the child was considered an adult, but what age was considered an adult during that era? The Wikipedia page of Sengoku Jidai did state the phrase "came of age" but didn't explain at what age one is considered adult.
There are no set ages for participating in combat. Generally speaking however, the first battle for a young samurai was in their early teens, roughly around 15 years old (opportunities permitting). Examples include Hōjō Ujiyasu at Ozawahara in 1530 (15), Takeda Nobukatsu at Temmokuzan in 1582 (15), and Date Masamune against the Sōma clan in 1581 (14).
Cases of samurai fighting at younger ages, such as Uesugi Kenshin in 1543 during the Echigo Civil War (13) or Kikkawa Motoharu at Koriyama in 1541 (11) are rarer, and probably limited to children of prominent families. General mobilisation orders, at least, were largely issued to able bodied men aged upwards of "fifteen". In the traditional East Asian system, that is equivalent to 14 years old in modern terms. Conversely, many notable samurai did not experience their first battles until about 20 years of age or even later, for example Chōsokabe Motochika at Nagahama in 1560 (21).
Participating in combat is somewhat linked to adulthood in that it was generally limited to "adults". In Medieval Japan, the coming of age event for boys was known as genpuku (
元服). This involved bringing the child before the family god, changing him into adult clothes, and giving him the haircut of grown men. The child also loses his childhood nickname and acquires a formal given name.
Upon completion of these steps, a child became an adult in the eyes of the law and customs. For example, Oda Nobunaga was born in 1534 and named Kippōshi (
吉法師). His genpuku was held in 1546 at Furuwatari, at which point he assumed his more familiar name of Kazusa no Suke Nobunaga.
However, there was also no set ages for genpuku during this period. While most boys probably held theirs between the ages of 12 to 16, there has also been extreme cases going as early as the age of six: Tokugawa Yoshinao was born in 1600, but held his genpuku rites in 1606. This is obviously wildly unrealistic for participating in combat, so adulthood isn't quite the same as going into battle.
There is no minimum age for a kid to be officially enlisted in a military unit without performing any duties and with the parent drawing his salary as a form of graft. No doubt babies have been officially enlisted.
Judging by several examples from significantly different times and places, it is possible for a child as young as five to be a soldier in some real sense, usually as a trainee.
The absolute minimum age to perform different types of combat actions is determined by the different ages at which different boys are large and strong enough to perform those different types actions reasonably well.
However, it is certainly possible for a soldier to perform military functions as part of a combat unit without fighting, and thus without needing to be capable of fighting. Drummer boys in western armies up to the late 19th century are a good example of young non combatant soldiers who sometimes performed non combatant military duties during combat and under enemy fire.
During the US Civil War drummers included men of all ages and boys of more ages - some of them younger ages - than would seem possible. The very few youngest musicians who enlisted -including some who served for long times and seemed reasonably successful and useful soldiers - enlisted as young as nine and eight.
It would certainly be rarer and rarer for samurai to fight in battle at ages younger and younger than the usual age range, since they would be less and less likely to be physically able, and trained well enough, to fight.
But participation in battles, campaigns, sieges, and other military operations has never been limited to those who fight in the strictest sense. Therefore it seems certain that some samurai boys (as well as commoner boys) participated in battles, campaigns, sieges, and other military operations before they were old enough or expected to fight.
At the very highest level Antoku-tenno never fought but was present at the Battle of Dan-no-ura.
You are applying Western mentality to Japan.
Samurai means "servant." A samurai's purpose is to serve his lord. Whether anyone fights, child or no, is up to the lord, not up to the child's father.
Likewise, the age at which a man might be accepted as a retainer to a lord is purely at the lord's discretion, so there is no fixed answer to your question. If the Lord orders a boy of four to carry a spear, then the four-year-old goes into battle and it matters not whether he is saka-iki or zangiri tafuto. It's that simple.