Yes the Palestinian Liberation Organization does have legitimate grievances with Israel. They both have legitimate grievances with each other, because they both have initiated an attack on each other at one point or another. There is no way that the Palestinian Liberation Organization does not have a legitimate grievance with Israel.
It has some as far as human rights issues go and the way Arab people are treated in general, and Palestinians in particular. However, Israel has a right to the land. The people living there do as well, but as citizens of the country they are in - Israel. They should be treated as equal, though, and they aren't always now.
No, I do not think that the reason that they hate the entire country of Israel, and all of the people in it is not legitimate at all, and is a very dumb reason. They simply hate them because of the religion that they practice, which is up to them.
No, the Palestine Liberation Organization does not have legitimate grievances against Israel, because it was made very clear to Palestine what land was Israel's, and the PLO just chooses not to accept it. Israel has a right to exist. Rather than acknowledge that right, PLO commits terrorism against Israel in an attempt to take over.
While many would argue that the Palestinian Liberation Organization does have legitimate grievances with Israel, I would have to say that ever if they are legitimate, they can't be satisfied, because they more than likely would have something to do with breaking apart Israel. Israel elected to recognize the PLO in 1993.