I am looking for the counterpart of the term "teenager" for those who are younger than 10 years old. A single word that means something like "elementary school student" is helpful as well.
I don't think you're going to find a single word--if there were one, education departments would use it. Across New York State Education Department documents, students in elementary school (usually kindergarten through 5th or 6th grade, depending on the district) are referred to as "elementary students."
The AAP calls them gradeschoolers.
Personally, I call them “school-aged children.”
Gradeschooler: 5-12 yrs.
10-12 is a tween: short form of tweenager (ˈtwēnˌājər)
a preteen or a young teenager. "the hot Nickelodeon show for tweenagers" -- Google
So perhaps you could coin pretween, meaning younger than a tween.
a few words come to mind, you could use...
"A human being below the age of puberty or below the legal age of majority."
You could also use minor http://legal-dictionary.thefreedictionary.com/minor
a person under the age of full legal responsibility. "the court would take account of the minor's wishes"