The "correct" preposition is just a matter of idiomatic usage...
We ask a question of someone we hope may answer it.
We put or pose a question to that person.
We also sometimes direct a question at someone, but that wouldn't really work in OP's context. It's far less common overall, and tends to be used in respect of a single question (or single series of questions).
It's worth pointing out that all the above variants involving prepositions are relatively uncommon compared to the simple version - normally, we just ask someone a question.
Although all of your examples illustrate exactly the same question, for clarity's sake we'll take your examples in order:
• 3 kilograms OF oranges or 3 kilograms oranges?
3 kilograms of oranges.
In natural speech, "of" would be included. You might see it omitted in a shopping list, but never in everyday speech or writing.
• 80% OF humidity or 80% humidity?
80% humidity
"Of" is omitted in common usage. The "of" is included, though, when we reverse the order of the terms. You might hear a meteorologist predict either: "Tomorrow, we will see humidity of 80%" or "Tomorrow we will see 80% humidity."
• At the spacing OF 3 meters or at the spacing 3 meters?
At the spacing of 3 meters
Here, normal usage will include "of." A list of instructions—for instance, to a carpenter—might read:
"Height 3 meters, Width 2 meters, Spacing 3 meters."
• 100 kg OF N or 100 kg N?
100 kg of N
See "oranges" above. In narrative use, we would include "of:" "Be sure to bring 100 kilos of N!" A list of constituents, though, might read:
200 kg Ammonium Perchloride
100 kg Nitrate of Gallodinium
150 kg Phosphorus
• 30 inches OF precipitation or 30 inches precipitation?
30 inches of precipitation
The "of" will almost always be included.
• 50 meters OF height or 50 meters high?
Either of these usages may be grammatically correct, depending upon context. "High" is an adjective, and "height" is a noun, though, so this example differs from the previous ones. In normal usage, though, you would almost never see "It has 50 meters of height." Instead, one would say either:
It is 50 meters high.
Or:
It has a height of 50 meters.
It's important to understand that you will frequently see technical or scientific writing, technical support documents, instructional manuals, and lists of things, which omit "of" for brevity's sake. There is no "rule" governing such usage except that "of" should only be omitted when its omission makes a statement unclear.
Best Answer
Along with your son's teacher's interpretation, there is another interpretation if you just look at the grammar.
When describing a tennis match, for example, you could say:
In this usage, against is missing but understood to exist.
So, if we take your sentence, how it could be interpreted is:
Again, if this were a tennis match, it would mean that you were on one side of the net and you had six (animated and tennis-racket wielding?) balls on the other side of the net.
Note that there's nothing wrong with the grammar of your sentence, simply that it doesn't imply with in the same sense that you want it to. Either it means you played the game six balls (as the teacher indicated) or that you played against six balls (which makes little sense).
Also note that even if you do explicitly use with, the meaning could be ambiguous. It can either mean playing alongside someone (John and Mary played bridge with Martha and James) or it could mean an item being the object of play (the children played with the toys).
In the case of we played with six balls, the meaning would be assumed from context— and the fact that balls aren't sentient and can't be partners, meaning they should be the object of play.