Gmail – Filter to move email to the “sent folder”

To search for email sent only to you and not to a mailing list, for example, you have to add the following to the search box (and then you can create a filter using that search; there is a link in the expanded search box that says "Create filter >>" or something):

To:me AND -*

This means it'll search for email sent to "me" (that's you) and not to everybody else (including you) (that's *), but the first part (i.e., To:me) overrides that.

Try not to treat Labels as Folders.

Messages in your Sent folder are just those your archive that you sent yourself (they don't need to have label whatsoever). When you add a label to them, they will appear in that label, and of course they will stay appearing in Sent because you sent them. (I testes it myself, outgoing mail with a label are showed when you open that label.

What you say about Move to Inbox; Inbox is just a label like another. The only 2 things special about it is that all new incoming messages get the Inbox label (except when you have set up a filter to let messages skip the Inbox) and that the Inbox label is your Gmail homepage.

I hope this helps you, if I did not answer your question then I didn't understand it, please make it more clear than and I'll retry.