I have not been able to replicate the issues that you are having, but here are some clarifications that should help:
1) Is it possible to have a single contact associated with multiple campaigns?
2) If so, how should I add a contact to a Campaign?
3) I've created Activities that are associated with both Customers/Leads/Contacts AND a particular Campaign but have yet to see these Contacts show up under the Contact tab in the Campaign.
Customers can only be associated with one Campaign in Method. Usually, this is to tell you what campaign the customer originated from. You may want to look at creating opportunities for customers. You can create more than one opportunity for each customer, and you can link them to separate campaigns. You can also create activities from within an opportunity to schedule calls and follow-ups. Those activities will automatically be linked to the campaign that is associated with that opportunity.
I'm wondering if I hit 'refresh' or something similar that may have reprocessed things? I'm not entirely confident that the list will update in real time as I add new contacts because of this, do you have any ideas why this might be?
Pay careful attention to the filter that you are using in Campaign screen. If you look under Contacts, for example, you might be filtering to show only your user's Leads & Contacts, or Customers & Contacts. In my screenshot I'm showing shared Customers and Contacts. Additionally, if you've made changes recently, try the refresh grid button.
Hope that makes things a bit clearer