The RB is not helping
Can gun crews fire inherent FP of 2 and fire their AT gun in the same First fire and Final fire Phases? Or just one or the other?
JR has pointed to the most important rule. The full explanation is something like this:
For a crew to fire a Gun or a SW, it must do so at the cost of its Inherent FP (A7.352). Thus if it has already fired its Inherent FP, it cannot fire any Gun/SW.
The main exceptions to this, is A7.353 and C5.6.
A7.353 allows the Crew to use its Inherent FP as SFF/FPF or as Final Fire vs adjacent enemies (regardless of whether it lost its Inherent FP due to firing it or the Gun), placing a Final Fire counter in all cases.
C5.6 allows the Crew to use its Gun for Intensive Fire as long as it is not marked with a Final/Intensive Fire.
Together, this gives you the following options, dependant of fire phase:
PFPh:
1. Fire either Inherent FP, or Gun (until rate is lost). Mark the crew/Gun with Prep counter.
2. Fire Gun as Intensive Fire (with no range restriction) once. Mark it with an IF counter.
Enemy MPh:
1. Fire either Inherent FP, or Gun (until rate is lost). Mark the crew/Gun with First Fire counter.
2a. Fire Gun as Intensive Fire (with no range restriction) once. Mark it with an IF counter and the crew/Gun with Final Fire counter.
...or....
2b. Fire Inherent FP as SFF (with range restriction of nearest KEU) once. Mark the crew/Gun with Final Fire counter.
3. Use FPF with crew (unlimited number of times) and/or IF as OVR Prevention (once). Mark Gun with No Fire if using OVR Prevention.
DFPh:
If not having fired in the enemy MPh:
1. Fire either Inherent FP, or Gun (until rate is lost). Mark the crew/Gun with Final Fire counter. No IF available.
If having a First Fire counter from the enemy MPh:
1. Fire either Inherent FP, or Gun as IF - vs adjacent/same-hex enemy units only. Mark the crew/Gun with Final Fire counter.
If having a Final Fire counter from the enemy MPh:
1. Fire NA.
AFPh (assuming no movement of Gun):
1. Fire either Inherent FP, or Gun (no rate possible unless using Opportunity fire). Mark crew/Gun with Prep fire
2. Fire Gun as Intensive Fire (with no range restriction) once, but only if using Opportunity Fire. Mark it with an IF counter.
...this is all written from memory, so I'm sure there is something missing, but it should be mostly correct.