Triviality of Yang Mills in $d>4$?

Good question. I am not aware of similar results for YM. The $\phi^4$ case uses correlation inequalities for ferromagnetic spin systems. Unfortunately, not many of those are known for gauge theories. YM is an example of model with non-Abelian group symmetry like $SU(N)$. Even for much simpler models with $O(N)$ symmetry like $N$-component $\phi^4$ or spherical spins, not much is known as far as correlation inequalities when $N\ge 3$.

Note that in Yang-Mills theory you can always scale out the coupling constant, $$ {\cal L}= \frac{1}{g^2} {\rm Tr}\left( F_{\mu\nu}F^{\mu\nu}\right) \hspace{1cm} F_{\mu\nu}= \partial_\mu A_\nu-\partial_\nu A_\mu +[A_\mu,A_\nu] $$ so there is no limit in which non-abelian YM theory reduces to a free field theory.