No, each cardinal is an ordinal, but not every ordinal is a cardinal.

However, for each ordinal k, there is the cardinal aleph_k. So in that sense, if there are uncountably many ordinals then there are uncountably many cardinals. And there are uncountably many ordinals in the sense that at least there is a set of ordinals that is uncountable. So there are uncountably many cardinals in the sense that at least there is a set of cardinals that is uncountable.

Right, proper classes do not have a cardinality.

There is no cardinality of the cardinals, but there are uncountably many cardinals.