Similar to Regeneration while dying in 4th edition, does a character continue to take ongoing damage when dropping to 0 hp?
Also, does he continue to make saving throws for all effects while dying?
Yes, a dying character still takes ongoing damage at the start of their turn and can make saving throws to end ongoing damage and other effects that allow a save at the end of their turn. They also get their death saving throw, of course. (Rules Compendium, page 224.)