Yep, すん is actually する, and putting な behind a plain form verb turns it into a negated command (huh, does that make sense? I want to say it is the same as “verb not!” in English)
For those contractions you might find this reference to be helpful?
ってこと is a contraction of ということ which is basically a quotation and a nominalizer, so it might mean something like “It says ‘Don’t worry’” or somesuch? (I don’t have the manga, sorry)
