Only one Software application can utilize an Audio Driver at any given time. You cannot 'share' the driver between two applications. This is true for most audio interfaces.
You are either going to 'accept' audio signal from software via FireWire OR send audio signal via FireWire to software, in regards to using just the mixer.
In these setups, you either want to have two different Audio Devices; 1 for audio playback to the mixer via analog cables, and 1 for audio output from the mixer to send to your stream service, assuming you're using 1 computer to handle it all. Performance will vary depending on computer processing power.
Otherwise, you will want 2 separate computers to handle playback and recording, with two separate audio devices handling each process. The mixer should be the central piece for accepting all inputs, then sending stream signal via firewire to software.
Any other 3rd party software used to route audio is not directly supported and would be up to you to configure and troubleshoot.