1st: If the assassin beats someone in combat using assassinate ability it says he MUST take a life. But what if that person has the Golden Idol?
I would interpret Assassin's ability as a change in the normal reward selection (Gold, Object or Life), so when using Assassinate, he can only pick Life. Golden Statue overrides that selection, forcing the winner to take it. I would apply that to the Assassin as well.
2nd: When can you drop and or pick up objects on the crown of command?
"A character may ditch any of his Followers or Objects at any
time by leaving them faceup in the space he occupies." (p. 16)
"Any time", meaning, well any time. Doesn't even have to be your turn. You just can't ditch and pick up the same card in one turn. If you are alone on the CoC, you can pick up any face up cards there.