it is her choice if she even wants to apologize to you
you betrayed her by digging through her emails and then telling other people about it
how would her affair play into custody anyway? It won't. I have never agreed with people who keep that crap...you are putting it "on the shelf" just like she is doing with OW.
your apology has nothing to do with her actions... your apology has to dowith yours and what brought your marrage to the point where she would feel like she needed to have an affair in order to have her emotional needs met