For the DNA piece - not explicitly Hannah, but it could show that Frank is the product of incest. Mitochondrial DNA is passed down through the maternal line - so you have the same mtDNA as your mother, her mother, etc. If Frank has the same mtDNA as Sam, then there would have to be incest going on as both of their mothers would come from a common maternal line. On its own, this wouldn't prove that Frank's mother is Hannah as opposed to Sam/Hannah's mother, grandmother, maternal aunt, etc. But, it would prove that there was incest.
There are also what are called long stretches of homozygosity - there is normal variation in unrelated people, and therefore variation in their offspring's DNA. However, individuals whose biological parents are closely related (siblings, first cousins, etc), you see stretches where instead of variability between the two copies of the DNA, they are identical. This is why rare recessive genetic disorders are more common in consanguineous families.
As for your first question - I think I would find this much more interesting if Sam never actually knew that Frank is his/Hannah's child. But, I bet the connection is through the Aunt that Frank mentioned - Aunt was friends with Hannah, and Aunt's sibling might have been trying for a child and been unable to (or whatever the story may be) and took the child, possibly without ever knowing the true paternity.