If your tractor has a jake and you need it, use it! Those signs of No Engine brake are for the straight piped old school tractors that you can hear a mile away. If that is why you are not using the jake. JMHO
In snow and ice you use your brakes ! That's what they are for, the key is not over heat them. I was trained using Light Steady Pressure of 10 PSI or less. That's because drums dissipate heat and 10 PSI or less is the magic number if how much heat they can dissipate without over heating. So you just leave your automatic in regular mode and don't turn on engine brake.
Jake slows the truck, not the trailer. On dry pavement, the trailer tends to stay behind the truck, but if the pavement is slick (like in rain/snow) the trailer may try to pass you. That is what they mean. I do not drive an automatic, but I typically apply very light, steady pressure on the brakes before engaging and throughout the use of the jake if its wet out.
I use manual mode going up or down any grade when I'm loaded heavy. When I'm light I run in automatic. You need to keep it wound up to maintain speed up a grade with a heavy load, and the automatics just don't seem to have that figured out yet. When it's wet jakes are okay to use, just decrease your speed a bit from what you would do on dry pavement. Snow or ice, cut your speed in half, grab a lower gear and don't use the jakes.
Yes. Manual mode is disabled in my truck but I have a work-around that's about 90% the same as manual mode when I really need it. On my truck you can force a downshift/upshift without switching to manual mode as long as doing that shift doesn't cause the RPMs to exceed/lug the engine. I would gently apply brakes and downshift to reach the speed that seems safe for conditions.
Just a tid-bit of advise when using the jakes when it’s on snow. Engage your inner axel in to maximize the jakes. ( note, only engage it when the truck is not slipping )