Why does copper gain electrons in redox reactions, when it instead has a tendency to lose them?
As you already said it, it only gains electrons when it's oxidized, not like a Cu(-II) would form or so. Therefore it's just regular redox-chemistry. Copper is somewhat noble, or at least it has a mediocre redox-potential. When dissolved in acids it only reacts with oxidizing ones like nitric acid. On the other hand if you have Cu(II) and add an iron nail, the redox-potential of iron is much lower making it a good reducing agent. But that's basically the same for every element. Once you have an oxidizing agent it will be oxidized, if you add a reducing agent it will be reduced. Gold would do the same, sodium as well if you used lithium to reduce although gold doesn't readily give off it's electrons and sodium will do it quite easily. It really depends on the other redox-system you add. 

You may also take a look at the redox-series:



Here the stronger reducing ones are on top (but depending on the artist it may also be the other way). Here it means the upper one will go from left to right if the lower one goes from right to left. To take an example:

Zn will form Zn(II) and Cu(II) will form Cu(0) in a system where both are mixed. But Cu will form Cu(II) and Pt(II) will form Pt(0) if those are mixed.