Should there be any reason to expect the cofinality to be a strictly increasing function? Perhaps naively, I have to admit that I don't remember my first thoughts on it right now.
Koenig's theorem assures us indeed that the continuum does not have cofinality $\omega$, from which we can deduce that $\aleph_\omega$ is never the cardinality of the continuum.
As for the second question, there is no reason to expect the continuum function to be continuous either (except the obvious similarity between the words). Indeed $2^n$ is finite, and so the supremum of all finite cardinals is $\aleph_0$ and not $2^{\aleph_0}$.