core: Simplify core initialization logic

Move all core initialization into Core::init(), including things
that were previously done in CoreApplication and MonoApplication.
Make several functions private and non-static that no longer need
to be called from the outside.

Allow to construct and destroy the core directly, rather than
relying on Core::instance() and Core::destroy(). Explicitly
create and destroy the instance in the application to better control
lifetime.

Ensure core state is saved and storage synced on destruction, instead
of relying on explicit calls (which were partially missing for mono).