Your second star doesn't make sense... it's the web interface, or IMAP, not both. At least, not directly, you'd have to have two windows open, one with your IMAP client, and one web browser. OR are you stating that those are two methods you access by? If so, it's very possible the IMAP client is moving messages or removing them. If it's set to store send messages locally, it would put them in a local file, and not into the sent items on the web interface. Unless you followed GMail's help pages' instructions for setting up IMAP perfectly.
Also, when you #3, delete it from the inbox... firstly, you don't need to delete it, it's better to archive, so it's around later... you have over 15G of space for mail. And Secondly, how are you deleting... if you use the trash can icon at the top, you're not deleting a message, you're deleting a thread; which includes your sent message. To delete a single message, you select the drop down triangle on the far right, and select "Delete this message".