Deterministic error reduction, state-of-the-art?
Doesn't van Melkebeek's lecture notes already give a $O(1/\delta)$ bound? The bound there is $\lambda$ at most $O(\sqrt{\delta})$ and we can get $\lambda = O(1/\sqrt{d})$ using existing constructions. 

In Dwork's lecture notes as well, the condition required is that the expansion be $C/\delta$ for some constant $C$ (looking at a points in distance c is essentially using powering to improve expansion). Which again can be obtained with degree $O(1/\delta)$.

Perhaps there is a lower bound of $\Omega(1/\delta)$ on number of runs required.