Can I access data on a current Time Capsule via the internet? If so, how does it work? Would I be able to make Time Machine Backups then as well? Does it appear as a disk in Finder?
Yes, if you sign in to your iCloud account in the TC preferences. Fire up AirPort utility and make sure your iCloud account info is in the "Back to my Mac" section of the "Base Station" tab.
Your TC disk will show up in the finder under "Shared" on any macs that are also signed into iCloud/BTMM.
Not sure if you'll be able to select it as a backup disk though; Time Machine preferences don't have a standard file chooser UI, it seems to demand either a local disk or a Time Capsule on a local network.