All six examples are correct.
1) implies that person A has lived ten years in country 1, no more and no less.
2) leaves it open that person A may have lived longer in county 1 -- "Person A has spent only four months out of 10 years in which she has lived in country 1, living in country 2, and for the other three years she has lived in country 1, she has spent over six months in country 2."
3) and 4) are interchangeable. The word "out" can be omitted with no change in meaning. The point here is that "Person A lived in country 1 for 10 years" specifies that there are 10 years in question, no more and no less -- and the word "the" in the second part of the sentence means that you are referring back to those same 10 years. In linguistics and semiotic, this is known as anaphora -- the use of an expression whose interpretation depends upon another preceding expression. (There is a somewhat stricter definition of anaphora in rhetoric, and what we have here does not quite meet that definition.)
By the way, if person A had spent a time measured in years in country 2, I would opt strongly for "Person A has spent only two of the 10 years in which she has lived in country 1, living in country 2." -- but only if the years spent in country 2 were full years and not 24 months scattered throughout the 10-year period, in which case we would still measure that time in months.
5) and 6) are also interchangeable. With 5), you could just as easily say "Of the 10 goals that were scored, Person A scored 4." Once again, the word "out" is redundant -- whether you keep it or not is entirely a stylistic choice!
As to the use of appositives, these are only of use if you have an alternate naming of one or more of the entities involved. For example, if you have been talking of a hypothetical case using person A, country 1 and country 2, and then wish to cite a concrete example, it would be quite valid to relate the hypothetical and the concrete by apposition, like this:
Person A, Arthur, has spent only four months of the 10 years in which she has lived in country 1, Switzerland, living in country 2, Italy.
As for the comment about using "during", yes, this is possible -- but I (as an Australian native speaker) would still opt for "of", and prefer this to "out of".