Been that way since Season of Dawn. Imo, I preferred the old thought of them being platonic brothers-in-arms, but I will admit that their relationship is handled pretty well since it isn't their only defining traits and is generally kept behind the main story.
Its an old Russian tradition for nurses to kiss their patients after a successful treatment. Kinda surprised Bungie even bothered putting this in TBH.
Yes, Saint and Osiris have been together for a long time now.