The answer in case (a) is also yes. In fact, a stronger assertion is true: there exist models of set theory in which every set is definable without parameters. Such models are called pointwise definable, and (as a first observation) are necessarily countable. A collection of results surrounding pointwise definable models of ZFC and GBC (in which every set and every class are definable without parameters) are presented in "Pointwise Definable Models of Set Theory," joint work by Joel Hamkins, David Linesky and me. Here's a link to Joel's blog post on the paper, which gives an overview & link to the paper itself: http://jdh.hamkins.org/pointwisedefinablemodelsofsettheory/

The answer in case 1. is also yes. In fact, a stronger assertion is true: there exist models of set theory in which every set is definable without parameters. Such models are called pointwise definable, and (as a first observation) are necessarily countable. A collection of results surrounding pointwise definable models of ZFC and GBC (in which every set and every class are definable without parameters) are presented in "Pointwise Definable Models of Set Theory," joint work by Joel Hamkins, David Linesky and me. Here's a link to Joel's blog post on the paper, which gives an overview & link to the paper itself: http://jdh.hamkins.org/pointwisedefinablemodelsofsettheory/