What actually happens when it "goes offline"? Have you checked the server logs? Your host sounds lazy and/or incompetent if their response is "I don't know".
Yes, this is the first thing that you should check your server logs.
Or your end, you might want to check your plug-ins as well.
So most probably a server issue to me and not Wordpress per se. And your saying that the host doesn't know what's going on? Lol, would advise you to switch hosting service. Do your own research and find a decent to good hosting that really gives support to their customer.