I may have a workaround for the continuous switching between accounts in firefox.
I'm not sure if it works, as I don't have hotmail or live email addresses.
The idea is as follows:
-Download
IEtab for firefox.
-Set it to open hotmail/live mail in IEtab by default.
Now hotmail should be opened in a firefox tab, using IE's rendering engine. So it should be (almost) like it opened in IE.
Please post back here if this worked out for you or not.