Does reduction/ oxidation take place separately at a half cell electrode?
In a (standard) Zn/Zn++ half cell, no oxidation (and no reduction) is taking place. A piece of zinc is sitting in a 1M solution of Zn++ ions (with their accompanying anions). It is not imaginary, and on a micro scale, an equilibrium is occurring (could be occurring!) where Zn + Zn++  --> Zn++ + Zn. No overall change occurs; this is a reference point. If it were not stable, some process would be taking place that degrades the half cell and it would be reacting itself away. In reality, half cells are only useful to the extent that they don't react themselves away until they are connected to the other half of the overall cell.

Same for Cu/Cu++. However, when the half cells are connected (thru a salt bridge so ions can flow, and a wire so electrons can flow), two processes occur: electrons leave the zinc (the zinc oxidizes) and electrons flow to the copper (reducing it). No "process" or reaction occurs in the half cells until they are connected. 

The reality of oxidation and reduction require both to occur simultaneously - WHEN THEY OCCUR - but we can speak of each process independently of the other. When electrons leave an atom (oxidation), they have to go somewhere (reduction); if electrons are not flowing (no actual reaction) there is no oxidation or reduction. The half cell potentials then define places from which electrons will flow, or locations to which they will flow.