When I have just sent a mail, it does not pop up in the sent tray until later. Why?
It’s not intuitive to watch the message I am writing in the received tray.
There may be some delay with sending emails, especially on a slow connection or if there are large attachments. What happens when you click Send, is the message moved to Outbox until it is sent, then it is moved to Sent. There may also be some delay on IMAP or Exchange connections because the Sent folder may not be synced in real-time like the Inbox.
I think what you are referring to with the message you are writing in your received tray (Inbox), is related to the Conversation mode. While you are composing an email, it is automatically saved as a draft. If it is in response to an email you received, the draft will be displayed in that conversation even if it has not yet been sent. It is not actually in the Inbox, just displayed as part of that conversation that is in the Inbox.
You can disable conversations in Menu > View > Conversation.