Zeta potential can be measured easily with dynamic light scatter based techniques, and of course TEM and SEM has to be done with a transmission and scanning electron microscopes respectively. Nevertheless, its possible that the chemical suppliers will have these data which they can provide to you (in case you have some issues with getting these characterization done). Although it is always a good idea to do the measurements yourself, since you can never really trust the suppliers data. Ion dissolution sounds like a simple aging experiment, where you age your sample (depending on the time scales of your experiments) and then isolate the supernatant (centrifugation?) and measure concentration of the Cu ions at time, t = 0, 1,2,3...hours/days/weeks.

Depending on the size of your nanoparticles and their colloidal stability, you could think of using a zetasizer (Malvern), here you can measure both size (in solution, hydrodynamic radius) and zetapotential at once using different cells. Particles smaller than 10 nm, however, are tedious using this technique. Electron microscopy can then be used to compare with the DLS measurement, however, might show a slightly different result. As you are talking about copper, normal AAS (Atomic Absorption Spectrometry) might do quite well to measure solution concentration.

Good luck!