Only the Onyx 1640i has more than a couple of Firewire output channels. With 16 in and 16 out the 1640i would do what you are wanting. I would generally discourage mixing organ and PA audio for a number of reasons, but that's not to say that it can't be a good solution in certain cases such as your need to feed an in house network.
One of the benefits will be the ability to minimize prelude and postlude crowd noise over the broadcast. I used to have a pair of congregation microphones both for recording singing and liturgy responses as well as to add a touch of reverb to broadcast channels when the sound was too monophonic and dry.
Ideally the organ would sound through its own speakers in the sanctuary, but there are no absolute rules about what can work well. The goal of organ sound and PA sound is usually somewhat at odds as one needs maximum sound dispersion and the other needs to preserve speech intelligibility with controlled dispersion for clarity. The trick may be to limit the speakers that the normal microphone inputs are routed to.