I'm probably missing something obvious, but it seems whenever I try to move the root bone of my character after frame 35, and add a keyframe, it's being ignored. I'll scrub along the timeline to the new keyframe, but the character doesn't move to the new position that I set.
You have an Action strip in the NLA Editor which is overriding/controlling the animation of the object.
Here I have added a keyframe on frame 54, but because the end of the Action Strip is at 35 (green timeline marker), the keyframe is ignored:
To solve this you can extend the End Frame property of the Action Strip in the Properties shelf on the right of the NLA Editor (N) to be as long as your animation:
Or, if you don't need to use the NLA Editor, you can delete the strip by selecting the Action Strip called 'Jump' (it will be yellow/orange when selected, as shown above), press Space, type "Delete Tracks" and press Enter.