I get that centrifugal force is what appears to force you out of a circular motion if you don't have the centripetal force required to stay in it. For example, a stone being whirled around on a string would go tangentially outward if let go due to its inertia and the fact that nothing's making it go in circular motion anymore.
But in the case of a frictionless object sliding radially outward on a frictionless rotating plate,
(1) Why is it radially outward and not tangential?
(2) Why does it move anyway; the surfaces in contact are frictionless; why doesn't it just stay there as the plate rotates, so that it doesn't move at all? Isn't that what inertia is?
(3) Generally, why does centrifugal force act outward on objects?