Mobile broadband is very touchy about signal strength and from what
you are saying it looks like the connection is being dropped due to
that. When Firefox goes south does Chrome also go ? BTW - Firefox will
enable the 'work off-line mode if it loses connection to the internet,
as will IE in Windows for that matter so it's not a 'Linux' thing !!

Also Right Click on the network icon and select 'Edit Connections'
then go to the Mobile Broadband section, click on the entry in there
and then on 'Edit' and tick the box 'Make Available to all users'.
You'll need to give your password but you should find the connection
works better now.