Almost every human child that has ever lived has learnt one or more languages without being taught explicit grammar rules. In most cases, they acquire complete competence over the grammar of the languages by age 6 (though if there is a standard language, they may or may not have learnt the grammar of that standard, as opposed to the grammar of a different dialect).
Until the last couple of hundred years, most people who learnt other languages after early childhood did so without being taught explicit grammar rules. The degree to which they were competent in the grammar of the language they learnt would vary a great deal, depending on many factors, such as their age, the closeness of the new language to one they already knew, their motivation, and so on.
So in answer to your last question: native speakers of a language generally have full but unconscious command of the grammar of their language; if they have not been taught about the grammar, they are likely to be unable to explain the rules they used automatically.
Note that I am talking entirely about speaking and hearing: reading and writing is a different question, as those are abilities which come only with a certain amount of intentional effort at learning.
A conclusion from this is that it is never necessary to teach somebody the formal grammar of a language for them to become able to communicate in it; but it may be helpful for some people, especially if it is important to them to master a standard language rather than just be able to make themselves understood.