Does the stock mail app log out from the mail-server? There is an email app provided by Android. 

Does this app keep the connection alive all the while? 

OR 

Does it login/logout from the mail server each time it polls for new messages? 

If the latter is true, how does one tell the app to log out from the mail?
About your concerns:

Connection on air

There is nothing that a log out at the end of every session would improve. If you don't use the email client, it won't transfer any data. And if you use it, it will query the server for new email etc. If the client would perform a log out at the end of every use, it just had establish a new connection again. Conclusion: Data related to the e-mail connection will be the same time on air if you don't use log out as when your e-mail client would log out every time. No gain here. Just make sure that you have enabled a secure encryption.

Bandwidth

You are right, that polling is the worst thing that could happen. But not from a bandwidth perspective. Polling is just a quick packet asking "Is there something new" and the server replying "No" or "Yes and here is the data". The data would be send later on anyways, so no big bandwidth consumption. It is more the battery life that should concern you, when you have a polling e-mail client on Android.

That said, trust the Google engineers to build a e-mail app that tries everything to reduce battery consumption. If you use gmail you are safe, because the email app will use push notifications for new e-mails. If you use an IMAP account it depends on the server if it support the IMAP IDLE extension. If you use POP3 you are doomed, because it's a plain polling protocol.