Audacity has a very nice noise cancellation filter. Is it possible, using JACK with ALSA, to pipe live audio through Audacity's noise cancellation filter?
No, for real time processing you would want to use Ardour
I just found http://www.penguinproducer.com/Blog/2012/01/ladspa-noise-removal/ It lists some plugins to use for noise cancellation
Works rather well!