Eulerian and Hamiltonian Graphs · Hard problem

The Möbius ladder $M_8$ has $8$ vertices $0,\ldots,7$ arranged in a cycle ($01,12,\ldots,70$) plus the $4$ "long" edges joining each vertex to the one $4$ steps away ($04,15,26,37$), so every vertex has degree $3$. By how much does the minimum degree fall short of the Dirac threshold $n/2$? Separately, what is the minimum number of new edges needed to add so the graph gains an Euler circuit? Report the SUM of these two numbers.