Completeness, approximability and exponential time results for counting problems with easy decision version

•The first TotP-complete problems under parsimonious reductions.•Hardness of approximation for these problems and some of their generalizations.•A family of hard-to-approximate instances for the complete problem Size-of-Subtree.•Exponential time hardness results for Size-of-Subtree under variants of the Exponential Time Hypothesis.•Approximation preserving reduction of the general #SAT to instances of it that are promised to be self-reducible with easy decision.