Can the "mutual independence" condition in the Lovász local lemma be weakened?
The Lopsided Lovasz Local Lemma relaxes the mutual independence condition to negative dependence. We assume we have events $A_1, \ldots, A_n$, with a lopsidependency graph $G$ defined on $[n]$ s.t. for every event $A_i$ and every $S \subseteq [n] \setminus \Gamma^+(i)$, 
$$
\Pr\left(A_i \mid \bigwedge_{j \in S}{\bar{A_j}}\right) \le \Pr(A_i).
$$
As you (sort of) suggest, the classical inductive proof of LLL works fine with the lopsidependency graph in place of the depdendency graph. This answers your question because the negative dependence condition is only weaker than the equality you wrote. See these notes by Vondrak for some applications.

The tightness of the constant $e$ (in the limit) was shown by Shearer, who characterized the smallest probabilities that guarantee a local lemma-type statement for any given dependency graph. There are fascinating connections with statistical mechanics, covered in a paper of Scott and Sokal.