I think question 1 uses the present perfect because no time is specified. "At any time up to now" is implied. I therefore think that the answer to the question can include both living and dead directors.
If the answer to question 1 were simply a list of names, it could include Burt Lancaster, even though he is dead, but if you wanted to make Burt Lancaster the subject of the verb "to win", you would have to say "Burt Lancaster won an Oscar". Though the question uses the present perfect, the answer does not necessarily do the same: "X [a living director] has won an Oscar" but "Y [a dead director] won an Oscar".
Question 2 - "Name five directors who won an Oscar" - suggests to me that Oscars are no longer awarded, but in practice, this is not the case, I think (I'm not absolutely sure because I don't follow this type of thing). If Oscars are still awarded, "Name five directors who have won an Oscar" is appropriate.