There are certainly options for connecting to Office 365 programmatically from a desktop application using the .NET Managed Client Object Model (Client OM). There are a number of posts on my blog (see signature line) detailing MS and community resources and additional information relating to using this approach to connect into Office 365 SharePoint Online including the upcoming Wave 15 version.
I'd suggest starting at http://dev.office.com to learn the new app model for Office and Office 365. Microsoft has been working very hard in the new release to align the Office 365 Web Apps with the desktop apps, including the development model. You may find that you don't even need to install software on the desktop to achieve the functionality you're hoping to build! If you build a killer app you can sell it directly through Microsoft's App Store.
Office 365 Home Premium is aimed at households and consumers and is not for commercial use - there's a quick comparison table here: en.wikipedia.org/.../Microsoft_Office_365