When I click on Computer properties, it says "System Type: 64-bit operating system" so I assumed my Office version would be 64-bit version. When I look in the program files folder, I can see that I have both the 32-bit version and 64-bit versions of Microsoft Office there. How do I check which one is actually being run? Does the auto updates check my version before feeding me an update and then gives me the appropriate version update?
Your system type is just the architecture of the operating system. Many (if not most) of the applications you likely use *are* 32-bit. Just because you have a 64-bit operating system doesn't mean everything that installs (Microsoft Office included) automatically installs the 64-bit version of itself - if it even exists. Want to know more about that, this article, "Choose the 32-bit or 64-bit version of Office" should be a good read for you.
If you want to know the true architecture of the Microsoft Office you have installed:
Open any Office application. In this example, we’re using Word 2010, but it works the same in the other office apps. Open the File menu, and then select Help.

On the right side, you’ll see information about your version of Office. It will show the suite version and included applications, as well as your precise version number and product ID.

Notice on the right of the version number it shows that this copy of Office is 32-bit. This is the most common version of Office 2010.

Or, if you have the 64 bit version of Office installed, you’ll see the 64-bit tag after your Office version number.

If you are being offered 32-bit updates - you likely (99% chance) need 32-bit updates.
So...
Questions:
- Do I need to install all the 32-bit updates?
- Do I need to click on each update individually and download each 64-bit version separately via the "more information" page?
- Yes - any that are offered to you automatically.
- No - if it is not offered to you automatically, don't try to manually install it.
64°F / 18°C