Oh boy, another internet oil thread!!!
Axles, gear oil. Something around 90wt. 75W90, 80W90,
Transfer case, factory recommendation is 50wt motor oil, which happens to be about the same as 90wt gear oil. Gear and motor oil weights have different scales, like temperature can be had in Celsius or Fahrenheit but both are in degrees. Some go hunting down the 50wt racing oil thinking it is the only thing that can go in there. Others just fill it with gear oil. To this day no one has ever had an issue that was related to using gear oil over motor oil. In fact gear oil is the recommended fill in all non-bronco Dana 20 transfer cases. For some odd reason Ford had this thing about using motor oil in manual transmissions up through the 70s.
These gears and boxes are fairly crude by todays standards. They will live with just about anything in them. The hypoid gear sets do need gear oil. And there are reports of people who didn't know better filling the transfer case with AFT and not having issues.