The cells were probably loose due to the wrapper not being uniform in thickness from brand to brand. You can take some clear scotch tape and put around the end that goes in the holder. That should add sufficient thickness to the loose ones.
About the soldering and copper. Corrosion is usually caused by differences in metals and metal interactions. So my guess would be that the solder you used may have something in it that is interacting with the copper. Or, when you soldered to the copper, you didn't get it hot enough to allow for the flux to properly clean the copper before the solder adhered to it. This causes impurities to be present and cause the metal interactions. I think if you got your copper bars hot enough during soldering, this would alleviate a lot of this from happening.
Now, if it's corroding due to oxidation, then that would still be under the aforementioned fix. Heat it up hot enough to allow for the flux to clean the metal, then the solder flows into where the flux was once at and then adheres to the copper. Essentially, the solder becomes a protective layer over the copper to keep it from oxidizing. Ie, tinned copper
Oh, and less I forget, nice looking pack
In those pics, it looks like you did a much better job of soldering to the copper wire. Of course, it's only 1 strand, not multiple strands like in the others.