A comment but I am not entitled. As regards the question at the end, the comment of Nate Eldredge points in the right direction---the result follows from the closed graph theorem since the convolution is continuous on the space of distributions.  Note however that we do not have a Fréechet space here. However it is a strict $LF$ space (Dieudonné and Schwartz) and Grothendieck showed in his thesis that the CGT holds in this context. As regards the first question, let me add that a necessary and sufficient condition for this to hold is that the sequence be bounded in the sense that it is uniformly bounded as well as all sequences obtained by successive differentiation.