The settings should not effect outgoing mail. Manual means that mail will only be downloaded to the Mail app when you open it or select the update icon within the app.
Push works with most IMAP and Exchange accounts. When the server receives a new email, it pushes it to the device.
Fetch is the device itself periodically contacting the server to check for and download email.
There is no way to keep Mail from downloading email when you open it. However, since you are on a GMail account you are using either IMAP or Exchange Active Sync to get your mail. This means that whatever you do in Mail, online, or another app will automatically sync to the other services. So if you you do something in Mail, it will change on the server immediately. When you do something on the server (either with the web interface or the GMail web app) it will sync with Mail as soon as it connects.
You can safely ignore email in the Mail app if you don't want to use it, though if you've opened it with unread mai on the serverl the red badge will appear until you either mark those emails read in Mail, or open it again after you've read or deleted them in another app.
If I've just confused you, ask away. I'm sure I can do better, maybe even get you all the way to bewildered.
