Frame jumping , simple python script to move the current frame pointer , once a keyframe is set you can use the buttons that move between keyframes.

keyframes: set your tool to keyframe ALL , that means everything in the scene gets keyframed at once.  See the pic , its the This Element / All elements radio buttons .

A parented item should follow its parent without specifically keyframing the child object.

The setting keyframe linear could be done by script. Look at Break Spline though, its commonly used when there is an abrupt stop or change in direction.

Learn to use the drop down selectors for body parts and props.