Don't be afraid to let the engine rev, both uphill and down. A roaring engine is a working engine, but it's made for it so long as the oil is good and the engine temperature is under control.
On the uphill, watch the temperature gauge. It should be fine if everything is working well. You are also heating the transmission and the differential, and I wish that they had gauges as well. Keep those fluids up to date, including the differentials.
On the down hill, hold the lower gear to keep the revs up. You might be able to coast down hill with the roaring engine for the whole mountainside. The important thing is that the brakes are stone cold. Keep back from whoever is in front, and use the brakes a bit every now and then if you need to. Don't ride them continuously, though.
Some mountain roads have "escape ramps". These are short exit ramps made of deep gravel intended to stop a runaway truck. Using one of those can damage your vehicle, especially if you have a trailer in tow. Think of it as a semi-controlled crash.
There are also slow vehicle lanes on the uphill. Here you'll find the big rigs doing 15 MPH with flashing lights.
Sometimes you can pass on the downhill in the oncoming uphill passing lane ( dashed center line), and I've seen crazy stuff here as a bunch of cars try to get ahead of slow trucks by passing at 90MPH downhill in the uphill lane.
Failure to downshift has been known to cause welded brake calipers. I saw this on an RV in a campground being repaired by a mobile mechanic.
If the trailer has electric brakes, you MUST use a brake controller. It's actually a legal requirement in many states, and failure to do so could in theory be used against you in court if you cause damage. Pay for a more expensive inertial-sensing model.