When I create a client side rule in Outlook, I can click "Run Rules Now", and it will apply my rules (which move mail from the inbox into subfolders) to all past mail items. However, this button doesn't exist for server side rules. Is there any way to run my server side rules so they will run on past, not just new, mail items?
Not really. Your best bet is probably to set up search folders to find mail items matching your criteria and then move the results as you wish.