> The complaint says they notified Amazon and GoDaddy, and apparently no take-down resulted, so I think DMCA does not apply.
I find that rather suspicious. If you file a proper DMCA request than the provider must, ASAP, take the content down - regardless of if the content is actually infringing. This is to protect their "safe harbor" status to make sure the government doesn't go after godaddy but rather the individual (rightfully so - godaddy doesn't and shouldn't control or monitor what users upload on their services). An example of it in use at github [1].
My sneaking suspicion is they either they didn't actually send a DMCA takedown request or they filed it improperly. Or perhaps the owners of the sites sent a counter-notice which puts godaddy in the clear and forces these individuals to sue the site owners (if they wish to have the content removed).
> Can a commercial service monitor a given email address and keep track of breaches affecting that address?
I think there is a difference between "here is a text box - enter your husband's email to check if he is a cheater" and "we will alert you privately if your email is ever leaked". I don't see any problems with the later.
[1] https://github.com/github/dmca