“hypothetical question”
Wife left Husband, husband found paper with wife’s email password on desk with the family computer. Husband accessed the email and discovered online communications with a male friend. These communications were sexual in nature. Can Husband use this information?
If the emails were stored on the family computer, yes.
It would be an online provider - such s hotmail, gmail or yahoo
I believe what Erin meant was this:
If the husband used the family computer (not the one at the library, or his work computer, etc, but the computer at home that everyone uses) to access the yahoo, google, etc email, then it is legally obtained.
What if she had already left the home?
If the computer had been used as a family computer, and was left in the home, then I believe it would still be considered the family computer. But, I would definitely defer to Erin. 
“Stored” in the family computer. does this mean it has to be in the cache memory? But what if the person wants to claim it was their computer when you put a keylogger on it? They can’t claim both fo their own purpose then claim it not being a family computer for a keylogger right?
I had this done to me, nothing was on there that I would care about, but it was used to change ALL passwords and he said he had a right to look cause he knew the passwords. But then added the fact it was “his” computer when filed a cyberstalking charge and included the keylogger as a reason.
Stored means that the email was already read and still accessible on the machine. Keyloggers are not illegal.