How can I list all user-owned functions?
The answer you are probably looking for is 

select * from user_objects where object_type = 'FUNCTION';


However this won't return a list of packaged functions. I'm not aware of anyway of returning these other than by querying user_objects for objects of type 'PACKAGE' and then describing each in turn. Someone with a developer hat on might be able to answer the packaged function part of the question.