Babalax, it is run by Colorado Bicycling Adventures.
Southerngrub - I have never done the tour but guests at various lodging businesses where I have worked over the years have all been very positive about the experience. While there will most certainly be road traffic, I have not heard of any accidents involving cyclists from this tour or otherwise.
I can only guess the Old Fall River Road tour happens in that span of time when the road first gets plowed of snow but before the road opens to vehicles. Maybe it is available other times, too, but as Babalax mentions, that would be an odd setup with that road being one way uphill. I have never noticed bikes heading downhill other than when the road is closed to vehicles.