Canon XF105 streaming
Looking at the capabilities of the XF105, there is no reason you could not simultaneously record to memory card and use the SDI or HDMI output of the camera.  It appears that you can get clean outputs on those feeds to avoid menus and such being present on the output.

The XF105 can't directly stream however, so you will need some type of device to create the stream.  You can either use a computer with a capture card or use a dedicated stream encoder that generates the stream directly.

For audio, if you have the ability to send an aux output from the mixing console, I would suggest using the XLR inputs on the XF105 in line mode to take a direct audio output feed from the sound board itself.  This way both your recording and the stream will have direct audio from the console.  If possible, doing a custom mix on an aux output is slightly preferable since ideal levels will vary slightly between the room vs a direct recording, but using room levels will produce sufficiently high quality results if you don't have the manpower to do both mixes simultaneously.