Pt 2: some models purposely leave the fan off until temps hit 60C, i.e. once you start gaming.
Pt 1: Core should be low when idle, but the GPU can help out on webpages so you need to be sure it's truly idle. Two screens may impact this, try it with one screen and idle, this was an ancient problem that i'd been told had been solved, but i'd never checked.
I just tried using one monitor and the clock speed dropped to 135MHz and the temperature is now 41c. Should I override the fan settings with MSI Afterburner to have it on even when idle at a low fan speed? I'm not sure whether having a 55c idle temperature is damaging the card over time, or whether having the fans on all the time will damage it either. Any info is appreciated.