Yes the add-in will load on Outlook desktop only if your mailbox is Exchange. I am not sure the add-ins work with a personal Microsoft account (e.g hotmail.com, outlook.com) and Office desktop. Keep in mind that the support of mail add-ins for personal account is relatively new (officially announced in march 2016).
To develop your add-in I suggest you to use a organizational account (i.e. Office365 account). Support for organizational account has been released in 2013 and is a much more mature solution for you to get started. You can have a free dev account at dev.office.com.
Edit: October 2016 personal account are supported now by "Personal mailbox accounts" aka outlook.com accounts
Let us keep in mind that a mail add-in is not installed on a mail client (desktop, OWA) but on the mail account. When you "install" an add-in you only set a xml file on your mailbox that basically says "there is a web add-in out there called X and that is served at this url...". Then, if you install add-in X on your mailbox [email protected], this add-in will load the add-in web iframes whether you are consulting your mailbox with Outlook Web Access (OWA) or Outlook Desktop.
To install the add-in for one user mailaccount, you have to use the Outlook Web Access. See this blog post. Note also that you can install the add-in for the whole organisation (Admin > Exchange Admin > Add-ins). Finally, Visual Studio is also able to push the manifest to your office 365 mailbox. To have this you need Office Development Tools.
For debugging in Outlook Desktop, I know two approaches.
- Use the Office Developement Tools with visual studio
and select your sample add-in project as 'Starting project' and in properties select 'Office Desktop'.

- Plain web development.
Personally, I do not use really much the Office Development Tools. I install manually my add-in with "https://localhost:XXXX" for url and I start the web project on IIS express to do the debugging. For stepping into the source code, I use the debugger of my browser (Chrome, Edge, IE etc.) when browsing in OWA. When I need to step into source code specifically on Desktop (For windows or for Mac) I use Vorlon.js see also this blog post.
My advice is also to test/debug frequently with IE 11 (still available in Win10!) and OWA. Indeed, the browser used internally by Office Desktop for the add-ins is IE.