If I understand your question correctly, the first part can be rephrased as: Is the system of all sets of a given cardinality a set or a proper class. It is a proper class already for singletons: Just notice that $x\mapsto\{x\}$ is a bijection between all sets and all singletons. Or, for a given cardinal $\varkappa$, you can use the bijection $x\mapsto \{x\}\times\kappa$ {x\}\times\varkappa$ between the class of all sets and a class consisting only of sets of cardinality $\varkappa$ (although not all of them). If singletons (sets of cardinality $\varkappa$) formed a set, then (by axiom schema of replacement) the class of all sets would be a set too, a contradiction.
(By the way, this is pretty much along the same lines as what Robin Chapman and Stefan Geschke wrote in their comments.)
