To make Outlook go to the Windows system tray when you minimize it...

Right click on the Outlook icon in the Notification tray on the right side of the Windows Task Bar and choose Hide when Minimized.

Tip: reopen it using this icon, not an Outlook shortcut, as the shortcut will open more windows. (You can add /recycle to the shortcut to re-use the open Outlook window.

To create an Outlook Contact group from a category in the Contacts folder ...

Apply the By Category view.
Select all the contacts in that category.
Choose Email (in the Communicate group).
Copy all the names from the To box on the resulting message.
Create a New Contact Group.
Click Members > From Address Book.
Paste the names copied in Step 4 to the Members box at the bottom.

To clear the Open Other User's Folder list, the Find a Contact list and other most recently used lists...

See Clear Outlook's Most Recently Used Lists

To troubleshoot Rules Wizard rules ...

If you can't figure out which rules are operating on messages, add an action to each rule to assign a different category. When you look at a message, its categories will tell you which rules fired.

To make undated tasks appear at the bottom of the list while dated tasks are placed in order at the top ...

There isn't a really satisfactory solution to this issue. The best I've been able to devise is giving undated tasks all the same date, very far in the future. This makes them pile up at the bottom of the task lists, while the older tasks are at the top.