yes, with quicktime you can select the audio source you wish to record along with the video...but only one audio source at a time.
one solution would be to get an audio interface. you'd plug the axe fx and your microphone into that and then select the audio interface as the audio source in quicktime. this doesn't solve the logic drummer issue though, so the best option may be to try Jack. Jack is an audio router that allows you to pipe audio from any source inside or outside your mac and send it anywhere you like. including multiple sources or destinations
http://jackaudio.org