Regarding the shutdown/restart, you might need to be a little more patient. During shutdown, Firefox performs some file clean ups such as finalizing history database writes, updating files listing open tabs/windows, and if you have set Firefox to clear history at shutdown, to do that cleaning. Firefox will kill its processes if this isn't completed within about 60 seconds. If you keep an eye on yours, are they lingering past that point?

Regarding the first session working, but the second and subsequent sessions not working, that is odd. I think there were one or two other reports of Firefox losing the ability to connect at some point during a session, but I can't recall anyone identifying the reason for it yet. Firefox does have a maximum number of connections it will make to a single server, but if you open a new tab and try a site that isn't open, the fact that Firefox may be blocked for already open sites should not affect other sites.