where is "He is" part of this sentence? could it as well be she is my child?
The "He is" part of the sentence is omitted, like so often. :)
It can be a "she" as well, of course.
Disagree. This is a wrong translation. For example: Who did it? Not my child.
It is not as much a wrong translation as it's another possibility. "Not my child" is not a complete sentence and needs some context.
Just as well you could ask: "Who is he?" and answer: "He is not my child."
You mean why is it not "I am not my child" (okay, this makes little sense) or "You are not my child"? Because those would require a verb. Remember that in identity sentences ("[Thing] is [quality]") only van and vannak get omitted. All other forms of "to be" are written out: "Férfi vagyok" - "I am male."
So "You are not my child" would be written as "Nem az én gyerekem vagy." (Unless you're speaking in formal terms, of course. But that would be a little odd with this sentence.)
They are pretty much the same, but "gyermek" is more "official", "gyerek" is more "colloquial". So, you will definitely see "gyermek" more on signs and such. Also, children may be addressed like that by someone who is not their parent, in a possessive structure. As in:
"Come, my children" - "Gyertek, gyermekeim".
"Come, children" - "Gyertek, gyerekek".
I think this is just a usage thing, not a grammar rule.
Also: "We are all God's children" would be expressed as "Mi mind Isten gyermekei vagyunk".
But, "A gyerekek a játszótéren.." - "The children on the playground..."
etc., those kind of everyday sentences usually use "gyerek".
I've sometimes wondered if there is a connection between gyere and gyerek. People often do say "Come here" to their children.
It's a nice way to memorise, but these words are probably not related, although the origin of neither of them is known. It is suspected that gyere derives from an earlier imperative form jerj, which then got affrigated and lost its final -j. The form jer, as a variant of gyere, can still be found in poems.