The outgoing mail saved as drafts issue, results when you are composing a message, Mail saves it to drafts folder on the server. You can select to save draft messages to On My Mac. You won't have access to your drafts on other devices with this option, but no more multiple drafts or a draft of a sent message.
In Gmail > Settings > Labels in your browser, select to sync the All Mail folder. In the past, many users selected not to sync this folder, but in Mavericks Mail, it needs to be enabled. Remember even sent messages are included in the All Mail folder.
Your problems are very similar to the ones in this discussion.