Fox co-owns BTN with the Big 10.
The Pac 12 owns its new network(s) outright; there is no involvement by FOX and ESPN, other than, as you mention, the new money they're pumping into the conference makes it possible for the network to be launched on the scale it's being launched.
ESPN and FOX, through their deals, get the primary choices of football and basketball games. The conference retains the rights (broadcast, digital, everything) to every other event that happens on a Pac 12 campus. But P12N is not, as you originally stated, a marriage between the three parties.