1 and 3. That Wikipedia link has the example 
$$\hat\delta^M= \frac{x+0.5\sqrt{n}}{n+\sqrt{n}}$$
for estimating $\theta$ in $Bin(n,\theta)$. It's not finite-sample efficient because that's one of the models where the MLE, $x/n$, *is* finite-sample efficient.

2. The local asymptotic minimax theorem says that an asymptotically efficient estimator is *locally* asymptotically minimax (over neighbourhoods of size $n^{-1/2}) for any bowl-shaped loss function, so if there's a counterexample here I think it would have to be a bit pathological.