Could be line problems, could be connectors getting dirty, seems strange that e-mails ok though. Try shutting everything down powering off at the mains, go round and re-plug all telephone lines, if you have any spare filters change them, then start everything up again and see if it improves, if no get them to do a line test for you.
Green Granite, sounds like the call centre script monkeys have messed with your mind.
Half of the things you list are things they use to buy themselves more time and make it sound like it's nothing to do with them.
If your computer is working and you can ping other devices on your LAN and you can ping your router, but you can't ping the outside world (either by DNS lookup or IP address). Check router connections and try rebooting the router to re-sync with the exchange. But after that, it's more to do with them having to pull their finger out and start testing things.
Unless you have made any significant infrastructure changes between the time when it was working and it suddenly stopped then you are wasting your time going round changing filters, unscrewing faceplates and all the other rubbish they spew out off their pointless scripts.