FPE for fox culling?

The .22 will do the job for good head shot placement but if your mainly just shooting fox and you don't mind the extra cost of pellets then I would go with the .25 cal because if you pull a shot or something the .25 is going to give you that extra bit of forgiveness.

Yep I agree with the above .22 and bigger would be better but your needs even though I believe your .177 would drop them on the spot with a PERFECT head shot. 

I've seen kill shots with a 177 but they were made at close yardage maybe 15 20 yards,
my NP 2 22 cal will punch holes in the side of a washing machine at 30 yards The NP2 is something like 250.00 

Know any other hunters? You could get some heart-shot carcasses and shoot them in the head at different ranges. Put it on YouTube and get famous.

What is humane?

https://www.youtube.com/watch?v=khrJwZLOn6s

Most any of the factory .25's will have ample fpe at 35 yards to humanely dispatch a fox with good shot placement. I have never shot a fox or fox sized animal with less than 20fpe at impact. I will say 20fpe at the point of impact makes for a humane kill.

As examples, I have a .22 that shoots jsb monsters to 48fpe and very accurate. I have used this gun to take raccoon, fox, and groundhogs up to 65 yards. Never had a further shot present itself with this particular gun but it is proven lethal at 65 yards.

I also have 2 .25's. One making 53fpe and the other 70fpe. Both have taken coyotes and smaller game out to and past 100 yards. Though the lesser powered gun has done a fine job, I use my 70fpe .25 when I know shots will be long. The extra fpe does make a difference since it still makes close to 40fpe at 100 yards.

FPE kills and you need to have enough energy at a given range, but shot placement will always trump all others when dealing with an airgun.

Before getting my 25 Marauder I ran a RWS 34 springer in 22. It pushes a 14 gr pellet to 700 fps and accounted for around 600+ fox squirrels in ten years. I went for hear lung shots mostly and found it to be only adequate. The 25 cal is deadly. It is easier to shoot well and hits so much harder. Head shots are easy now. The one issue with the 25 is pass through shots are the norm and I have to only shoot when I have a back stop. My 25 is running 40 fpe.

I wouldn't suggest anything but head shots with less than a 25 cal, even then to be truthful I would prefer a 30 or bigger for a body shot.

what is a clean ethical shot?..... a shot that kills the animal in the quickest manner balanced with the ability to perform that shot with confidence and repetitiveness.

Before I start my bow season hunt I shoot over a 100 times with my bow in practice, I try my hunting tips and find any difference in field points and hunting points, I shoot 3d targets marked with vitals at different angles elevations and light. I shoot from stands and blinds to see if my possistion will change my aim 

Ability is just as important if not more so than the caliber our power of the weapon