[Trexmaster]

Apparently God Rod will /always/ give you the full benefit of the effect, even if you don't have a staff weapon or you have Priest as your sub-class. (As in your FAI will /always/ be added to your spell power with Curate/Priest spells. I only tested this with Shine Ray and Holy Spark, but, I'd imagine it'd go for all the other spells.)

I tested this numerous times to confirm.

Dark Water also doesn't seem to affect the FAI stat at all (at least via Black Bubble--the item made tiles weren't tested.)

[Blissey]

Tested this with Holy Spark with a staff on my off-hand. This is happening. Whenever you equip a staff on your off-hand, however, with Priest is main- or sub-class, your damage will instead decrease. If you don't have a staff equipped at all, God Rod's effect still applies. Wow, what a doozy.

Without Staff:
--- ROUND 0 ---
Doctor Mavis Haemin's turn.
Jammer's turn.
--- ROUND 1 ---
Doctor Mavis Haemin's turn.
Doctor Mavis Haemin's turn.
Jammer takes 90 Light damage.
Jammer takes 90 Lightning damage.
The battle has ended.
Gained 1 Mura.
You consume your emergency food supply.

With Staff:
--- ROUND 0 ---
Doctor Mavis Haemin's turn.
Doctor Mavis Haemin's turn.
Jammer takes 79 Light damage.
Jammer takes 79 Lightning damage.
The battle has ended.

[Neus]

God Rod has been corrected.

Dark Water was also not working (I'm not sure how this hasn't gotten reported until now...)