Sounds like fun - I'd start by riding on quiet roads, and I'd fit studded tyres to all four wheels. You can swap back to non-studded come summer.
Look if your ebike function has a limiter function. You don't want full power else wheel slip.
My trailer always gets a skunk stripe on the front, from the rooster tail of the rear wheel. BUT its got a "full" mudguard. If you want to avoid throwing mud and roadwater at your kid's faces, look for a rear mudguard that goes a long way around the wheel, and hang a leather dangler/mudflap on the end. These are more often used on the front wheel, but when towing will keep the trailer cleaner.
Comms - I have a bunch of handheld radios, and use them for all sorts of weird things (eg dog training) It would be hard to talk to the load when riding, so you could hook a radio inside for them and one on your collar. The inside one may work with Vox mode (transmit on voice) so you can hear their chatter :)
Most important will be braking - your method may need to change, and if that's not enough you may need to fit your trailer with a brake.
Brake 80% with the rear brake, and only in a straight line else trailer will want to come around you. If the trailer does come around, you need room to stop braking and apply pedal power to get back in line, which needs road room.
Brake early when you need to stop.
If this doesn't help your braking you need to make the trailer brake the bike through the hitch, using some kind of brake on the trailer's wheels.
If you get a lot of snow, then you could fit skis. Either change both the trailer's wheels and the bike's front wheel for short skis or somehow strap skis to the bottom of the wheel and let the length of ski prevent rotation.
For drive, fit a chain to the bike's rear wheel. This would be a significant amount of work and only useful on snow/ice.