Configuring CNC axis

I'm new to CNC but currently making a 5-axis wood router. I need to be able to address 3 faces of a cube and I'm making a machine able to do this, however I have no knowledge of CNC software.
I am quite a competent programmer and adequate machine builder but I need to make sure the machine I make can be supported by available software.

The problem I have is that the rotating axis of the spindle will not be linear and will not have full rotation. Can software cater for this either from formulae (i.e. given equation for path) or from physical setting (i.e. from micrometer measurements)?

Any help or advice would be appreciated, thanks! :)

LinuxCNC will almost certainly handle the kinematics and you may want to use CNC-Toolkit, which is a free program for generating 5-axis toolpaths. You may have to do some programming to get linuxCNC to run the machine as it's not linear, but maybe not as there are a few existing things which may do what you need. Not linear and not full rotation implies you're making a hexapod (which linuxCNC supports directly), or just generally deriving the rotary motion from a linear actuator, i.e. ballscrew?