They are being totally unreasonable, they should give you a chance to have look at your site. Honestly, once this issue is resolved, ditch them and go with a decent host.
As for setting up your sites on a local machine, I'm afraid if you have never done that, you'll spend more time and precious hair on your head trying to that. If you are running a business, it would be penny wise and dollar foolish to have the sites down even a minute than trying to ask for free help.
I can help you setup the WAMP on Skype if needed, so if you need help, just give me shout here or just click on my name for skype and phone numbers.
Happy New Year and good luck.