Did he die through falling on his sword?
Malbim (here and Shmuel I, 31:4): No, he became only Goses (close to death). The Amaleki killed him, like he himself told David (Shmuel II, 1:10). It says that he died, for he was mortally wounded. Radak (31:4) #1 - "Mes" is like "b'Tzeis Nafshah Ki Mesah" (Bereishis 35:18).
Radak (Shmuel I, 31:4) #2: Yes. The Amaleki lied; he already found him dead on his sword. He thought that he will find grace in David's eyes by saying that he killed him.
Was Sha'ul permitted to kill himself?
Radak (Shmuel I, 31:2): Yes. Even though it says "v'Ach Es Dimchem l'Nafshoseichem Edrosh" (Bereishis 9:5), i.e. I will demand (punishment) if you kill yourselves, and similarly "mi'Yad Kol Chayah Edreshenu umi'Yad Ish Achiv", even so he did not sin, for he knew that his end is to die in war, for so Shmuel told him, and he saw that the archers found him and he could not escape. It was better that he kill himself, and the Arelim will not make a mockery of him. Chazal (Bereishis Rabah 34:13) expounded, perhaps "Ach Es Dimchem l'Nafshoseichem Edrosh" includes the case of Sha'ul! "Ach" excludes this.
