If driven with the cable disconnected or not adjusted correctly then you'll burn up the clutches in the transmission.
The cable controls the increase in line pressure as you increase throttle to prevent the clutches slipping, and it also controls the shift points.
I'd be tempted to get a new cable hooked up and adjusted correctly, then drive it and see what happens. If it slips then it will need a rebuild.
If you want to do a bit of investigation then drop the pan off and see if there's a lot of friction material inside. If so, plan a rebuild.
For the back axle, depends if you have a posi/lsd. If you don't, any 80w90 or similar gear oil will be fine. If you have a positive, use one with the additives in, I prefer the Titan one from Opie.
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My '80 progress thread - hugger orange!
Now with LQ9 power - forged pistons, LS2 rods, 236/242, .621/.612, flowed and milled 317 heads. MS3X
4L80e, 12 bolt, 4.11, Moroso brute strength posi, steel shafts.
Subframe connectors, g-braces, solid body bushes, carbon hood and front bumper, custom door cards, Autometer gauges.