From chemical abundance analysis of stars in the Sagittarius dwarf spheroidal galaxy (Sgr), we conclude that the {alpha}-element deficiencies cannot be due to the Type Ia supernova (SN Ia) time-delay scenario of Tinsley. Instead, the evidence points to low [{alpha}/Fe] ratios resulting from an initial mass function (IMF) deficient in the highest mass stars. The critical evidence is the 0.4 dex deficiency of [O/Fe], [Mg/Fe], and other hydrostatic elements, contrasting with the normal trend of r-process [Eu/Fe]_r_ with [Fe/H]. Supporting evidence comes from the hydrostatic element (O, Mg, Na, Al, Cu) [X/Fe] ratios, which are inconsistent with iron added to the Milky Way (MW) disk trends. Also, the ratio of hydrostatic to explosive (Si, Ca, Ti) element abundances suggests a relatively top-light IMF. Abundance similarities with the LMC, Fornax, and IC 1613 suggest that their {alpha}-element deficiencies also resulted from IMFs lacking the most massive SNe II. The top-light IMF, as well as the normal trend of r-process [Eu/Fe]_r_ with [Fe/H] in Sgr, indicates that massive SNe II (>~30M_{sun}_) are not major sources of r-process elements. High [La/Y] ratios, consistent with leaky-box chemical evolution, are confirmed but ~0.3 dex larger than theoretical asymptotic giant branch (AGB) predictions. This suggests that a substantial increase in the theoretical ^13^C pocket in low-mass AGB stars is required. Sgr has the lowest [Rb/Zr] ratios known, consistent with pollution by low-mass (<~2M_{sun}_) AGB stars near [Fe/H]=-0.6, likely resulting from leaky-box chemical evolution. The [Cu/O] trends in Sgr and the MW suggest that Cu yields increase with both metallicity and stellar mass, as expected from Cu production by the weak s-process in massive stars. Finally, we present an updated hyperfine splitting line list, an abundance analysis of Arcturus, and further develop our error analysis formalism.