You can use 80/90 hypoid type gear oil if you wish but you can also use 15W40, 20W50 or straight 30W in the trans-axle. It's up to you.
It's a good idea to flush the old milky oil out of there by pouring in a quart of diesel fuel and driving the tractor around for ten minutes or so and then draining the rear end once more. Then put in a quart of any motor oil, drive the tractor for another 10 minutes and drain that out. Refill with 3 quarts of your choice of oil.
The Tiller chain case gets packed with regular chassis grease. Only you can decide whether to change it or not. You must dismantle the entire tiller to do this. New seals on the output shaft would be needed and so would a new #50 chain. Check the top and bottom sprockets for wear and tear. This is more of a winter-time project. These tillers do last a long time and breakdowns are rare.
Just be sure you take lots of photos of how the tines are bolted on BEFORE you dismantle everything.