For the bridge, I like them to do a very brief pause on it. Basically, as I'm walking up I ask my horse to put the head down and really look at it. (I have been known with young horses to put a little bit of grain in a trail right down the center of it, and let them lip it up as they are going over on occasion). They need to be nice, slow, and responsive while going over it, and a very brief pause allows the judge to know that the horse is waiting on you. You can also put a pole on top of the bridge as something else for them to pay attention to.
To help them pick up their feet, at a walk I like to have them 1/2 way over a pole and then suddenly change their direction. I will also just lay the poles out in a big huge mess with no pattern to it and let the horse just pick their way through it.
I don't side pass through the gate. I kinda leg yield them up to the correct location, stop, and unhook the gate. Then back up a step or two so their nose will get by the end, then it is a kinda hindend pivot/leg yield to open the gate, walk through, turn on the forehand to face the opposite direction on the other side of the gate, then leg yield them back over to the correct spot and close it. Then leg yield them a couple of steps away from the gate, and off you go.