Can I search through someones email if they left their email logged in at my house?

Probably.  If someone leaves papers open in front of you, they would generally be considered to have waived whatever rights to privacy they may have had.  This is an analogous situation.  However, I can't guarantee that a judge will see it the same way.  However, most court proceedings allow for discovery, where you can demand that the other side produce any relevant documents.  Serve your adversary with a document request.  If they don't comply, and you can prove it, you may win your case just on that basis.