It's been a long time since I played around with the Axiom settings and I have an older model so things may be a little different for you.
You set the Axiom Preset to 17 via the + and - keys in the numeric pad area (labeled 'Snapshot').
The display will show P17 or whatever.
Each of these is a different setup and Preset 17 is one that sends controller data the way most things that I use like it.
You can probably see the list of differences in the Axiom manual.
Enigma is the free M-Audio MIDI set up software that let's you do as much damage as you like to the MIDI controller stuff. You can download for free from the M-Audio site.
What you are calling channel is a bit confusing. MIDI has 16 channels and these are usually reserved one for each instrument (and 10 is usually a Drum machine).
The actual name is Controller number - 'Continuous Controller' (CC) to be more precise - and there is a preset list of many of these - 1 is the Mod Wheel, 4 is a foot controller, etc. But you are pretty free to change these as you like. However, many things are already set up to recognize things like Mod Wheel data, so if you do change them, do so at your own peril.
Enigma lets you make your own presets of the Axiom settings - you can change zones, split the keyboard, change velocity curves, lots of stuff that I once played with but now forget completely.
I usually use a DAW for this and just use the Axiom as a keyboard with a few switches. Mod Wheel and Foot controller are my main external controllers.