the CC's (continuous controllers) are sent from the midi controller via USB just like note data. Each knob per patch on the axiom will send out some form of CC, it can be in several different format (usually 0-128). The mapping will be stored on the Axiom and Reason will have its own default mapping for each instrument also, the 2 just need to match. Reason actually sends the right map to my Remote automatically on startup, so I dont really have to deal with it
You need to setup you controller as a "generic controller" (check help or the manual for "control surfaces")
There will be a default map for each instrument. Axiom has presets for reason built in (check manual page 48) so just change to those when you want all the controls mapped accordingly (presets 7-12). You can also overide the default map by either right clicking on the control, selecting "edit overide mapping", checking "learn from control surface" and simply moving that control, or you can look at all the controls by going to option/keyboard control edit and double clicking on the control you want to override (it will turn into a spinning lightning bolt) and then moving the control on the axiom
As far as controlling other hardware/software. You will need to look up the documentation for each devices midi mapping. Some are static so you will need to make a patch for the axiom manually (use the software) and some have a learn function like reason so you can leave the controller alone and program the soft synth instead. Midi learn IMHO is useful really when you just want to tweak a couple of parameters and building a whole learned map is kind of a pain.
As far as bank and program changes, this is done via "Bank MSB/LSB" which uses two controllers to define the bank/patch. Here is a good resource since I don't feel like writing a super long explanation:
http://users.chariot.net.au/~gmarts/midimsg.htm
The controller is just a midi device and is seen as such my all programs that utilize midi so it is effectually a generic midi port as far as everything is concerned.