I changed the wording of your question slightly to make it more clear. The title of Muzychuk's paper is 'On skew Hadamard difference sets'.

What is n? If it is $k-\lambda$, then your $N_v$ and $N_{v,n}$ appear to be the same. I am not aware of results for cyclic groups, but Muzychuk has constructed exponentially many equivalence classes of difference sets in certain elementary abelian groups.

Both STS(13)'s are written out on page 29 of the Handbook of Combinatorial Designs. And no, the automorphism group of the second design is small, of order $6$. (It is easily seen that no non-trivial design with block size $k$ can have a $k$-transitive automorphism group.)

The 'group' operation defined on orthogonal orthomorphisms after the second question is non-associative: $(\alpha \ast \beta) \ast \gamma = \beta^{-1}\alpha\gamma$ while $\alpha\ast(\beta\ast\gamma) = \alpha^{-1}\beta^{-1}\gamma$, where I assume the operation on the right is the usual composition of permutations.

If one wanted a group covered by seven subgroups, surely the elementary abelian group of order 8 is an easier example?