When I get extra energy in a casket or as a reward in play, do I have to use it at once?

That depends if your total energy is below or above the full limit, i.e.110 for regular or 150 if you have the extended bar.

If your energy is below full, then the auto refill will only continue to fill until it's full and no more. So when you return to your game later, any bonus will be indistinguishable from your usual full bar. Therefore, you should not let that bonus energy sit there; use it up.

But if your energy is above the full bar limit, that energy will remain there until your return, no more, no less. This is how players stockpile energy for competitions, by collecting energy from landmarks, coffee energy, etc. until it is well above the full bar. They won't get auto refill energy, but they can accumulate large amount of energy for particular events.