In OS X when Calendar.app is closed I get notifications about events; how can I have this work in Mail.app? While Mail is closed I want to get new email notifications.
You need the Mail app running. Unlike Calendar events, checking for emails is not something that can run locally. A client application need to connect with your mail server and retrieve your e-mails.
Sure there could be some background process that does that without the need of the Mail app, but it would not make much sense; in the end you would still have a process doing the checking.
Something that may make the Mail app less intrusive to you desktops' organization etc is having it been launched automatically and hidden at startup.
System preferences>Users>Login Items> + > Mail >Check off the hidden button.