"Antreten" doesn't mean "to compete" directly. The meaning in this context is more like "to enter (a competition)", "to show up (for a match)", "to step up (to the opponent)" or "to report (for taking an exam)". It might help to note that having a group of soldiers to form up is called "antreten" (or "antreten lassen") as well.
But you're right in that "antreten" is often used to describe the competition that happens afterwards as well. This is especially the case when "antreten" is used with the preposition "gegen", like
Er tritt im Halbfinale gegen den Champion aus dem letzten Jahr an.
He steps up in the semifinal to last year's champion.
He competes in the semifinal against last year's champion.
"Konkurrieren" doesn't fit this meaning of "to compete" really well, either. This verb is more commonly used in a more abstract sense, for example
Die beiden Unternehmen konkurrieren um die gleichen Kunden.
The two companies are competing for the same customers.
For a more concrete competition like a sports match you can use several verbs. With regard to your examples
Children between the ages of 9 and 12 will be allowed to compete in the tournament.
Kinder im Alter von 9 bis 12 können am Turnier teilnehmen.
Here, "teilnehmen" means that the kids can enter the tournament, they can participate in it etc.
Team A played with a lot of energy, team B didn't even try to compete.
Team A hat mit viel Energie gespielt, Team B hat nicht einmal versucht mitzuhalten.
Here, "mithalten" means something along the lines of "to keep up" (with team A).
Tomorrow, the German team has to compete against the Italian team.
Morgen muss die deutsche Mannschaft gegen die italienische Mannschaft antreten.
Here, we can used "antreten" as explained above. The German team has to step up to the Italien team (and, implicitly, compete against them in the match).
If you want to go for the exact meaning of "to compete" in this context, you can for example use "sich messen (mit jemandem)" (literally "to measure up oneself (against somebody else)". But this wouldn't sound too natural in most cases.
In diesem Spiel können sich die Kinder mit ihren Altersgenossen messen.
In this game, the kids can measure up themselves against their contemporaries.