What’s the connection between שאלה and שאול (Sheol and Shaul) from root ש.א.ל.?

The word sheol and its various forms appear close to seventy times in the Bible. Sheol’s literal meaning is “grave.” Interestingly, Ibn Ezra to Gen. 37:35 criticizes the Christian Vulgate for translating sheol in that verse into the Latin infernus (“inferno”), because Ibn Ezra maintains that sheol literally means grave. However, Rashi (there) explains that although the plain meaning of sheol is “grave,” exegetically it can refer to the post-mortem purgatory of the soul. The Malbim writes that sheol literally means a deep pit from which it is impossible to get out. This applies to both a “grave” and gehinnom. Rabbi Shlomo Pappenheim of Breslau (1740–1814) explains that the root of the word sheol is SHIN-LAMMED which denotes something “thrown away” or “negated.” That meaning extends to the grave because death marks the onset of a plane of existence which is “away” from the realm of the living. My friend Rabbi Tzvi Matisyahu Abrahams takes a more exhortative approach in his book Root Connections in the Torah. He writes (p. 274): “The grave is called sheol because at the time when we will be placed into the ground, there will be a big question (sheilah) mark hanging over our heads as to where we will be headed.” Alternatively, we propose another connection between the word sheilah and sheol: sheol literally means “deep pit”, which serves as a metaphor for one who asks a question. A question creates a burning need for an answer. Just as a deep pit serves as a portal to the empty abyss, a question leaves a gaping hole in the questioner’s mind that begs to be filled.