How can I move sent messages automatically from Gmail (IMAP) to my local PST file?

You could most probably access your Gmail ‘Sent’ folder from within Outlook. Can’t you just drag them onto your local ‘Sent’ folder from there?

The detail of your question seems unclear. If you are using POP on your Outlook client and don’t have the “leave messages on server” option checked in GMAIL, all messages received in your GMAIL folder and downloaded successfully to Outlook will be deleted from your GMAIL inbox.

IF you are using IMAP on your Outlook client and have GMAIL configured for this correctly, you need to set-up Outlook to sync with GMAIL (e.g. you see the GMAIL inbox within Outlook like any other account) and create a rule that moves the individual emails from the synced IMAP folder to another folder physically located on your system (e.g. a local folder or inbox). When creating a rule in Outlook skip the standard rules and create a custom rule. It’s not difficult but read the text for each option carefully as sometimes Outlook won’t let you do things in a manner that is as direct as you’d like.