Why does it jump so awkwardly when the part doesn't fit and it has to readjust?
The cloth receptacle is being compressed like a spring. It seems like when the robot thought the object was put away (or some safety feedback mechanism of receiving too much force feedback activates), it "relaxes" its actuators and goes limp, say. Then the stored force from the spring is released. What we're seeing with the jump is the robot rebalancing itself in order to remain upright.
Not a chance. The compartments are clearly just hanging there and not under the kind of tension that could cause something like that.