Strictly said: Bunny is twice in your sentence.
La kuniklo havas ses kuniklidojn.
Unfortunately not; "ido" does not mean "child," and so "idoj" cannot be translated as "children."
Yes, it would be okay, when "children" is understood to refer to the family relation, not to the age of the younger rabbits.
Commonly, in English "offspring" is used both for plural and singular. It's similar to how the word "sheep" has only one form, used for both singular and plural. So, though the Esperanto can have "ido" and "idoj", the English is "offspring" in both cases.