May a child who has reached the age of chinnuch eat meat during the nine days? Is there a difference if it is Friday evening dinner assuming the child eats before Shabbat begins?
Rav Shmuel Kamenetsky in קובץ הלכות - בין המצרים, פרק ט הלכה ב-ה writes:
1 - Up to around five years old - there are no restrictions.
2 - Erev Shabbos, if a child is eating his Shabbos meal before Shabbos, even meat is permitted. However, if he is eating again with the family at night, he should not have meat during the Friday daytime meal.
3 - Even during the week, bird meat [chicken] is permitted even for those children that are of chinuch age.
This article from Ohr Sameach Rabbi Yirmiyahu Ullman, says:
The custom is to refrain from eating meat and poultry or drinking wine and grape juice during the nine days. This also pertains to children.
(I assume that he is referring to children above the age of chinuch.)
A very comprehensive answer is brought by Rabbi Simcha Bunim Cohen in his Children in Halacha
On p.187 he starts to delineate the laws of eating meat and chicken during the Nine Days. As far as the first part of your question about just stam being able to eat meat/poultry in the nine days, he asserts that healthy children above the age of three years old should also refrain from eating meat and poultry during this period. He notes (in the footnotes) the Rivevos Efraim, cheilek alef, siman 371 which brings the opinion of Rav Shlomo Zalman Braun (the author of She'arim Metzuyanim Ba-Halachah) which writes:
דקטן שאינו יודע להתאבל אסור בבשר ויין כשהוא בריא ונ"ל דעד ג' שנים אינו בחזקת בריא דכמו שהוא מסוכן
That a minor who doesn't know how to mourn is forbidden to have meat and wine when he is healthy, and it appears to me that until 3 years old one is not considered to be 'healthy' and it is like he is in danger (i.e. if he does not partake of meat etc in his diet)
Rav Simcha Bunim Cohen adds:
Therefore, adults may not feed children any of the aforementioned food items. If the child is unhealthy, or will not eat any other food, one is permitted to feed the child meat or chicken.
Then with regards to the second part of your question about feeding a child meat on erev shabbos, he adds (p.188):
If a child is eating a Shabbos meal Friday afternoon (i.e. after plag hamincha see p.58), one is permitted to feed the child meat or chicken.