It shouldn't happen if your weight is over the center of the bike, unless the trailer is really, really heavy. The exception would be while climbing a steep hill, when a more moderate weight could be a problem. In that situation you might want to position your weight more over the handlebars
That's true with a normal bike. With weight on the handlebars it's unlikely unless the trailer is loaded wrong. My folding bike has crazy geometry though. The seat is really far back and the riding position is very upright. It puts most of the weight just in front of the rear wheel.
In most states a bicycle is considered a vehicle, meaning you must ride in the same lane and going in the same direction as cars, signal your turns and obey stop lights and signs. When I see someone ride a bike against traffic or on a sidewalk, running a stop sign, etc., I have two nearly simultaneous thoughts: (1) he should get a ticket; (2) he's probably going to get hit by a car. It seems to me either result is undesirable from a survival standpoint.
It's like that here too but there are areas of the city where riding on the sidewalk is allowed (typically where the sidewalk is really wide and/or the traffic is really dangerous). I pick my route where I can stay on bike paths, sidewalks where cycling is permitted, or one way streets...I like not having to deal with oncomming traffic.
One trick I use when traffic is heavy is I'll stay right even when I want to turn left. I'll stop at the cross walk, flip my bike 90 degrees and wait by the sidewalk with all of the peds for the light to change before going down the cross street. Then I don't have to change lanes and I don't have to cross oncoming traffic. Good return on safety for a small investment of waiting time.
I also like lights and reflective stuff (I haven't bought a vest yet though). Drivers might complain that flashing lights are distracting...but I say that's the point