For your first question, yes, it absolutely jumps up under acceleration. You can see this in many videos, as the car is so damn powerful, and its suspension soft enough, it almost wants to do wheelies.
For the second question, AFAIK, batteries are more efficient when they're hotter because there is less impedance and internal resistance in the battery, thus allowing the lithium ions to flow more easily. IIRC, a cold battery is like cold oil, where the oil(lithium ions) is more sluggish when cold, and more free flowing when warm.