In Opera Mail, sent messages show up in "All Messages/Sent". Unsent messages show up in "All Messages/Outbox".
For both views, click the "settings for this view" icon on the message list toolbar for the view and make sure "show sent" is checked.
For IMAP, sent messages will only be local and not copied to the server unless you set an IMAP sent folder (to "Sent Items" for example) on the "IMAP" tab in the account's properties. However, some mail providers' SMTP servers will automatically copy a sent message to the sent folder on the server. Gmail is one example. In the case that a provider auto-copies sent messages, whether you can keep an IMAP sent folder set in Opera or not to avoid an extra copy, depends on the mail server. For Gmail IMAP, it'll be fine.
When an IMAP sent folder is set, "All Messages/Sent" will show messages that are in the IMAP sent folder.
For drafts, they show up in "All Messages/Drafts". For IMAP, Opera doesn't support an IMAP drafts folder so drafts are never copied to the server unless you manually do it by right-clicking a draft item in the "All Messages/Drafts" list, choosing "copy", opening the drafts IMAP folder under the you@example.com access point, right-clicking in the message list and choosing "paste".
Views under "All Messages" show messages from all accounts. You can right-click in the mail panel, goto "show messages from" and select a certain account if you want to show only messages in that account. It resets to "all accounts" on startup though.