Please see my last paragraph. If you adjust Elkies' example to have constant column sum, it fails to have large inverse. The inverse of the circulant has all entries less than one in absolute value. Indeed, being a multiple of a stochastic matrix, we would expect small column sums in the inverse.

They exist for $n=5,7,9$. In fact, resolvable such designs exist for odd $n$ up to $9$. A general construction seems elusive at the moment, but I wonder if conference matrices can be used.

I'm not sure how I missed the following for $c = 1/2$: Take a complete graph of even order $n$ and remove a spanning disjoint union of two stars, say centered at vertices $x$ and $y$. Remove a few more edges for $K_3$-divisibility. Then (1) the average degree is near $n$; (2) the minimum degree is $n/2$ (when the two stars are balanced); and (3) edge $e = \{x,y\}$ belongs to no triangle. Can $c$ be pushed even higher than $1/2$?

That's right, although I suspect the OP knows this already. Regarding naming, I think co-covering is not a bad term. But design theory already has a proliferation of special terms, so it's maybe best to complement parameters as stated above.

I just meant that in most cases the stabilizer is trivial. That is what I thought you meant by "generic".