But I am missing the reason why the electrons flow from the zinc to the copper in the first place. Even if they oxidize and reduce simultaneously, what initially causes the copper to be reduced and at the same time that the zinc to be oxidized. I don't understand the initial driving forces.
For the second case whatever happens there is between Cu and Zn, at the same oxidation level, a bIt different from what we are considering - Cu2+ and Zn.I think I understand. Just to be sure, if I touched solid copper to solid zinc instead of connecting them with a conducting wire, would the copper oxidize some of the zinc atoms?
I don't think so.
Your first question is really what drives any chemical reaction, or any ionic one with simple electron transfer at least. That is part of general chemistry. Some atoms hold on to electrons or give them away more than others. All tied with the explanations of the systematics of the periodic table. The elements are all different so one will tend to donate electrons to another, the big question is which to which. You can study that even without electrochemistry. Zn will donate electrons to Cu2+ giving Cu and Zn2+. You can easily find many you-tube etc. videos showing this happen and red copper precipitating out of blue copper sulfate solutions in which there some plates of zinc. (I only don't quote any one because those I saw were sufficient but not very good.)
The driving force you ask after is already there in thse reactions. Electrochemistry just has the nice addition that through diffusion of ions in the liqiid and the miracle of metallic conditivity you can couple one chemical reaction at one point to another at another, and also in the process extract useful work, like light up a lightbulb.