When animating a jump, should the characters feet stay on the grid and the actual up and down motion be handled by the game engine itself?
It depends. If you’re using legacy, it should stay on the grid. In legacy animation, your Unity scripts are responsible for moving the character. You can use products like SAGE to do this in a very nice, easy way, and it’s on sale today only for 50% off.
If you’re using Mecanim, you should take advantage of root motion by moving the character’s feet off the grid in your 3D modeling/animation software. If you don’t, you can script root motion using curves in Unity Pro.