False theory of how a battery works
For a moment I would suggest you to forget about electrons,their flow ,batteries and stuff. Assuming that you have the basic intuitions regarding the principles of Newtonian mechanics and work done and all I would like to proceed in a different way. Suppose you are standing now on a ground and you have a ball in front of you lying on the ground. It doesn't lift itself up in the air, which is a common observation we all have made by now. But when you wish to lift it up to some height,be it h, then you expend your ATPs and chemical energies in your muscle to lift the ball slowly to that height h , and in doing so you did a work exactly equalling mgh ( neglecting air drag) . Now you take a tube of some arbitrary shape ( just to give a proper intuition) such that the ball is at its one end and the other end meets the ground. Then you leave the ball and it cleverly curves through the tube and gets back to ground level. This is exactly what we find in a battery system. It pumps electrons, against an opposing conservative field and then allows them to fall back. Here ,there's no man power but the chemicals do the trick by getting indulged into some redox reactions. Now ,if you think of those questions from this perspective hope you will be able to answer them efficiently.