I'd say a minimum of 18 inches wide for a cage, as they are lively birds. For an all round cage I'd say 18 x 12 x 12 rectangle would be fine, as always anything bigger would always be welcome.
My finches had always been fed seed, a few weeks ago I decided to try the small finch pellets and mixed them half and half with their seed. Mine pick out the pellet and eat it instead of the seed. I'm not sure what brand it is, I'll look next time I buy some, but they love it.