You can save a character with all animations, sprite and visibility settings as a Project file but not as a saved Character file.
What you can do with a saved character is include animations in the form of an Action Menu. This allows you to include with the character a set of animations that can be applied by right-clicking the character and choosing Action Menu followed by the animation you want. This way you don't need to search the Content Manager to find the animations you need each time.
For help setting up an Action Menu, please click HERE