(1) https is used for all my logins, none with http. In fact this is enforced at the top level of my domain under Apps.
(2) The account you see is in fact my primary gmail account, under Google Apps. Any othe r email accounts I have elsewhere are forwarded to this account. Other such accounts are required =by many of us who have and services for years and have only relatively recently opened Google Apps services.
It is true that I have logged into into another google account. Google Apps users do not have access to logging into google news, search, or even the new google voice under chat. It is best and correct to consider Google Apps (even paid Premier services like mine under my own domain) as being for Users -- not Accounts per se. For these other services a separate google "Account" is required. So, for example from an Account user google News I have tried went to that Account's gmail just to look at my Google Voice.
It seems that Lastpass in fact fails to handle multiple accounts on Chrome. I think it handled those OK on Firefox, but I have not used that in some time.
Lastpass has failed to handle the login into my Google Apps, as I said, even after I cleaned all Chrome caches and cookies, and even cleared Lastpass of all "google". When I first went to the cleaned browser Lastpass asked if I wanted to save is email sitehttps://mail.google.com/a/mydomain.com
, but it still does not work.