Synchronous rectification in IGBT
Addressing your questions in order:


The IGBT will not be impacted in any way. IGBTs do not conduct current in reverse; this is why they must be coupled with a separate diode [1]. Even with the IGBT on, its free-wheeling (also called 'anti-parallel') diode will conduct the current. MOSFETs, on the other hand, can conduct in both directions. In the MOSFET, the current can reverse-conduct through either its channel or its body diode, depending on the current, channel resistance and body diode drop. The body diode is inherently built into the MOSFET. 
It will indeed be turned on, but no current would flow since the collector would be floating. An IGBT, like a MOSFET, turns on gradually. It begins to turn on when the gate voltage exceeds a threshold, relative to the emitter. 
Yes to both. The ratings you mention (10A, 600V) are the maximum ratings. You can use the IGBT with 25V bus and 500mA, no problem. 


[1] See an example of a TO-247 device which is effectively an IGBT and diode co-packaged: http://www.power-eetimes.com/imf/c/eyJtYXNrIjoiNDMyeDI4OCJ9/images/01-edit-photo-uploads/2014/2014-06-june/onsar2662_fig-1.jpg.