Can I get notifications with Mail App if App is closed?

You might possibly be able to do something with fetchmail. You or a wise geek would have to study it well to figure out how to accomplish what you want.

 

Do you mean close or quit? Why not just close it / hide it when you aren't using it? It'll be unnecessarily difficult to try and fetch mail with the application quit. If you use Gmail, you could use a menubar app that notifies when you have new mail. Then you could open Apple Mail to read and deal with it. But I don't see the point of quitting Mail at all if it's your main client.

 

No, there is not. Plus it defeats the entire purpose of having Mail closed. You could use GrowlMail and close Mail's window, which would have the effect you seem to be looking for.

 

There's no problem with leaving apps like Mail open all the time. It's not using significant resources just to check for new mail every minute.