Outlook – How to configure that sent mail stays on the original folder of the reply and not in sent items

I don't think you can automate that, since rules are designed to run before email is delivered, not before sent mail is processed.

The only other options are to manually trigger the rule (as you've already stated) or create a macro that does the move, and assign the macro to a button on the toolbar. Although that would be easier to initiate, it would be much harder to design the macro to do what you want.

I was trying to figure this out just now. On the sent folder, I changed the settings so that messages would show as conversations. This way, I can easily see which messages I've sent that do not have any response as there will only be 1 message in the conversation so there won't be a conversation tree.