Death before the fall
Whether physical death existed before Adam's disobedience, or entered the world with it.
Genesis 2:16-17
16Yahweh God commanded the man, saying, “You may freely eat of every tree of the garden;
17but you shall not eat of the tree of the knowledge of good and evil; for in the day that you eat of it, you will surely die.”
Romans 5:12
12Therefore as sin entered into the world through one man, and death through sin; so death passed to all men, because all sinned.
What each tradition says
Rabbinic / Midrashic
Some assign death to Adam’s first sin in partaking of the forbidden fruit, according to Tanna debe Eliyahu R. v. This view attributes death to that first sin.
Jewish Encyclopedia, Death, Views and Customs Concerning (Jewish Encyclopedia, 1901-1906) — Opening paragraph on the cause of death (first documented view)
Rabbinic / Midrashic
The Jewish Encyclopedia records another view: death was ordained at creation. Adam’s sin did not begin death; it merely hastened it.
Jewish Encyclopedia, Death, Views and Customs Concerning (Jewish Encyclopedia, 1901-1906) — Opening paragraph on the cause of death (second documented view)
Patristic
Augustine says Christ did not bring death Himself; He found it already here, propagated from the curse of the first man. This death of ours, which originated in sin, Christ took upon Himself and hung on the tree.
Augustine of Hippo, Expositions on the Book of Psalms (trans. J. E. Tweed), in Nicene and Post-Nicene Fathers, ser. 1, vol. 8 — Exposition on Psalm 38, §25 — the curse of the first man; the death He took upon Himself
Catholic
Thomas Aquinas cites Romans 5:12, which says that death came into the world by sin. From this he concludes that man was immortal before sin.
Thomas Aquinas, Summa Theologiae, Prima Pars (Fathers of the English Dominican Province trans.) — I q. 97, a. 1, 'On the contrary'
Orthodox
Gregory of Nyssa describes the grace of immortality falling away to mortality. Marriage then preserves the human race by means of descendants.
Gregory of Nyssa, On the Making of Man (NPNF series II, vol. V, trans. H. A. Wilson) — Ch. XVII (the grace of immortality fallen away to mortality)
Reformed / Protestant
No sourced position found. Checked Calvin's Genesis commentary (King trans., 1847 scan) for a statement on Gen 2:17 and death before the fall; the scan's Gen 2:17 exposition was not retrievable, and no other fetchable PD Reformed text on the question was obtained within this run.
Evangelical
Spurgeon describes the first pair eating the forbidden fruit, after which pride withers their paradise and blasts their Eden. They go out to till the ground from which they were taken and to bring forth children who are sons of toil and sorrow.
C. H. Spurgeon, The New Park Street Pulpit, vol. II (1856): Sermon 97, 'Pride and Humility' (Prov. 18:12) — Sermon 97, 'Pride and Humility' (the fall of the first pair)