Yes you do. If you go for the Business Premium subscription however you will also get a desktop version, so you can work offline. As soon as you go online your desktop version will sync all your files with your online version. Bear in mind the fact that if you have a fairly late version of Office already it will work with Office 365 - we're talking about Office 2010 onwards. When you sign up for Office 365 you'll get a little Microsoft present called Office desktop setup which does everything for you, including setting up the syncing with your current MS Office software. It uninstalls itself when its job is done.
You create your file storage structure the way you like it. The files are yours, you can store them offline if you like.
You can import information to Office 365 from your current version of Outlook.
I have been looking at obtaining Office 365, but looking on the Amazon website that is showing it currently at £48.32/annum there are a lot of adverse comment about it. Looking on the Microsoft website the price is £59.99/annum. I also found a reference to the fact that you are able to download from third parties. Could the problem that are being commented on are because the purchase is through a third party and the third party are cocking it up?