I have personally never used the subscription options, but I'll attempt to answer things that I can either look at the admin pages on, or ascertain by how things should work; I've also asked someone who is running the feature for some clarification.
I would also recommend that you setup a demo group so that you can see the items in action yourself.
First - Do administrators get notified when someone starts or cancels a subscription?
Third - Is there a "Thank You" message that is automatically sent when someone signs up for a subscription, and if so, can it be edited?
Fourth - If we set it up as a non-recurring subscription, will the subscriber be notified when the subscription is about to run out?
A PT should be started between both the user and the admin when a subscription has been started (think of the "Welcome" PT, only with the admin being included in the discussion); this should also inclue when a user has been added to a group, removed from a group, subscription expires, etc. These items are covered by the "subscriptions" language file; if you create a demo group you can get a better understanding of these strings so that you can make the options more clear for your users.
Second - We need to know our members real names and street addresses. Is there a way to automatically request this information when they subscribe?
I'm not sure that a user has an option to add data to a subscription; you could dedicate a custom profile field for this, and remove it from being displayed in the member profile (you'd also have to add the field to the member management section of the control panel, or have someone well versed in the coding update it for you).
Isaac: for REAL NAMES on the subscriptions, the admin processing the transaction gets the real name and address through either the CHECK PAYMENT they receive -- its on the check & requested in the "include your real name, user name, and make checks payable to AWESOME WEBSITE." language file. for PAYPAL, the transaction in your paypal account will list their email address and their paying names (real name) and payment address.
Isaac: As for storing the real names within UBBT, there is no field for such thing, unless you use one of the CUSTOM1, CUSTOM2, CUSTOM3....etc, to store that information, and allow the users to create it themselves (PUBLIC). The other idea, is to just add the real name to the "PRIVATE USER COMMENTS" where only the admins and user moderators can fully edit that field.