Transfer case: ATF. ATF+3 or ATF+4 is fine, it is happy with any ATF actually.
Transmission. Your AX-5 transmission will be happy with a synthetic 10W-30 engine oil or a 75W-90 GL-4 gear lube like Redline MT-90. Do NOT use a 75W-90 GL-5 gear lube, most are bad for the transmission. To put your mind at ease on the two seemingly vastly different viscosities I have recommended, 10W-30 engine oil is roughly the same actual viscosity of 75W-90 gear lube, they are measured differently. The manufacturer of the transmission, Aisin, added synthetic 10W-30 as an acceptable lube since it is so hard to find a 75W-90 GL-4 gear lube locally.
Axles. Use only a GL-5 gear lube in your axles. Use lower viscosities like 70W-80, 75W-90 for most needs, heavier viscosities like 80W-120, 85W-140 if you tow a trailer. I prefer a NON-synthetic for the axles as there is a lot of evidence to say a non-synthetic gear lube helps the gears to run cooler. The exact viscosity is not a big deal in the axles.
Engine, it's hard to beat 10W-30 except in extremely cold conditions where 5W-30 is a good choice. For gnarly cold like Alaska or Buffalo NY, a 0W-30 is good in uber cold condtions.