First, does the new Gyre XF250 work with the new Apex Controller?
Not at this time
If so, how does it connect?
See previous answer
Do I need to use an IceCap Module?
If you want to connect a gyre to an APEX controller, yes.
Or can the new advanced controller connect to the Apex controller?
Not at this time.
If the advanced controller connects to Apex, how does it connect and is there an advantage in using the new advanced controller over the IceCap Module?
See previous answer.