This worked for me with Outlook 2010.
1) In the left hand navigation pane, click on (highlight) the .pst file where the "missing" emails should be showing. Or, if the emails are just "missing" from a folder or subfolder, click on that folder or subfolder.
2) Click on the "View" tab on the top of the Outlook window.
3) On the far left side of the ribbon in the "Current View" section, click on "Change View."
4) There should be 4 options:
Hide Messages
Compact
Single
Preview
5) "Hide Messages" is probably the option presently applied, so click on the option you would prefer instead, and your emails should appear in your email window in the format you have just selected.
6) If "Hide Messages" does not appear as an option, click on any option other than the one presently applied and let the drop down box disappear. Your emails should appear in the email window in the format you have just selected. Then click on "Change View" again and choose your original format (assuming that this format is still what you want), or choose another option altogether. Your emails should now appear in the email window in the original or newly-selected format.
If you right-click a conversation in the conversations view, you can go to Move > Always Move Messages in This Conversation.
I have not figured out how to make this apply automatically, i.e. all new emails move to the conversation's folder. If someone knows how to do that, please post a new answer.
Best Answer
You can either use Outlook Archiving or you could add rules which route certain or all incoming mails to your personal folders.
The Archiving feature is time-oriented. All mails elder than a defined number of days/months get moved out of the inbox to some archive folder. This can be done a structure-preserving way, i.a. the folder/sub-folder struture of your online folder is reflected to the archive.
The rules can be added using Outlook's rule assistant. A "rule" selects incoming mails according to their subject, sender, addressee and other fields. The action part of the rule can copy, move or delete mails. It is also possible to spawn scripts or external programs.