Full subshells will be more stable than half-filled subshells, which will in turn be more stable than other partially filled subshells. In this case, the s orbital is more stable with 2 electrons than with just one, and so it takes more energy to remove an electron from the more stable subshell.
The phenomenon is more clear with full, half-filled, and partially filled p-orbitals. Take a look at this graph of ionization energies of different elements as one moves across a period:
http://4.bp.blogspot.com/_OdZZiyNKEoU/SmETmcVPFjI/AAAAAAAAAd4/H8aeFVmpa7E/s320/Periodicity+-+IE.jpgNotice how the ionization energy drops after adding a fourth electron to a p orbital; that's mainly due to electron repulsion.
As for the second question, the ionization energy has to do with electrons in the s orbitals, because that's where the first electron will be removed. I don't know the exact reason why Cu would be bigger than Ni, but I can guess the atomic radius has to do with the electrons in the d orbitals, which are probably farther away from the nucleus than the electrons in the s orbital because they are worse at shielding than s electrons (and consequently more penetrating).