Well, I wouldn't define Gmail as "social media" either, but Social Pro does. It offers support for multiple Gmail accounts, so the developer is aspiring to attract customers who want support for webmail accounts.
Both Yahoo and Hotmail can be managed through desktop mail clients, or at least, they used to be able to be managed that way. if they've changed their terms of service, you would be correct on your second point, but I'm not certain about that.