Is the English adjective "whole" genealogically related in any way to the adjective "holos", which means "whole" in Koine (and possibly other varieties of Greek; I'm not sure), and has a similar pronunciation? If so, how are they related?
They sound very similar but are actually unrelated.
The Proto-Indo-European root of whole is *koilas. From etymonline:
O.E. hal "entire, unhurt, healthy," from P.Gmc. *khailaz "undamaged" (cf. O.S. hel, O.N. heill, O.Fris. hal, M.Du. hiel, Du. heel, O.H.G., Ger. heil "salvation, welfare"), from PIE *koilas (cf. O.S.C. celu "whole, complete;" see health). The spelling with wh- developed early 15c.
Greek holos is related to English safe and several other words, including Catholic.
late 13c., "uninjured, unharmed," from O.Fr. sauf, from L. salvus "uninjured, healthy, safe," related to salus "good health," saluber "healthful," all from PIE *solwos from base *sol- "whole" (cf. L. solidus "solid," Skt. sarvah "uninjured, intact, whole," Avestan haurva- "uninjured, intact," O.Pers. haruva-, Gk. holos "whole"). Meaning "not exposed to danger" is attested from late 14c.; of actions, etc., "free from risk," first recorded 1580s. Safe-conduct (late 13c.) is from O.Fr. sauf-conduit (13c.).