Being logged off after a preset time when the connection is not being
actively used is a function of all dial-up accounts and whilst there
are artificial ways of keeping the connection open it's against the
T&C of any dial-up provider - it can lead to termination of the
service if you persists after being warned about it !

The reason is due to the way that dial-up works - inactive connections
prevent others from dialing in so they are terminated to give everyone
a fair share of the service. You might argue that there are fewer
users these days but the provider may not see it that way and might
well have enough users to make this still an issue for them. Also bear
in mind that dial-up equipment is old and getting to the stage where
repairs probably aren't viable, so it may well be that the ISP has
fewer workable connections too !!

As for Firefox - that again is likely due to the dial-up type of
connection and I don't think there's any way to override it. Basically
when Firefox starts up it doesn't *see* the internet connection
properly as it's too slow !

The advice would be to switch to broadband but that might not be
viable for you ( cost / location ) and if so then these annoyances of
dial-up will just be something that have to be borne. Many of us here
will shudder as they are reminded of their time on dial-up