When leading a horse, and especially when you are teaching a horse to lead correctly, you want to make it clear to the horse that, as you lead, the horse is never to go further than having the point of their nose parallel to the point of your shoulder.
In order to teach this, I will simply lead the horse forward a few steps, and abruptly stop. I make my own body cues very clear to the horse at this stage by leaning forward a little while I am walking forward and by being abrupt and standing very erect when I stop. Keep the lead rope slack, both while you are walking and when you are stopping. We want the horse to learn to pay attention to us and our body cues; not just to learn to be dragged around with a rope. If (and when) the horse does not stop when I stop, or if the horse walks at a faster pace than me and gets her nose beyond my shoulder, I will rather abruptly (but not violently, as that will likely create problems rather than solve them) take the slack out of the lead rope to pull on the halter and back the horse up. The horse will probably still not back up, or will be very sluggish about it so I will use the tail end of the lead rope as a quirt on the horse’s chest to create some backwards momentum and back the horse up several steps. I will do this every single time the horse’s nose passes my shoulder. After a while, I will begin to have the horse also back up on a loose lead rope whenever I back up, and I will pull on the lead rope less and less, going straight to using my body position, and the lead rope as a quirt on the horse's chest rather than making the horse dull in the face. The horse must learn to read my body rather than learn to be pulled on all the time.
When leading the horse through the gate, I will make sure that the horse learns to patiently be led rather than franticly rush past me. I will do the same thing here at the gate that we have already been doing: lead a few steps, stop, back up, and wait and be petted. If I know the horse has anxiety problems like this while walking through a gate, I’ll make sure that, for a month or so, I will make it a habit never to just walk through a gate; but to mix it up by walking partway in, then stopping, backing out and being petted at a standstill. Horses are creatures of habit, always looking for the next step, so sometimes a horse that rushes at the gate is not really even trying to be rude and is not really even scared; but just wants to do what he knows is coming next. Well then, if we make being backed up or stopped and petted in the gateway a habit, then the horse will think that what is coming next is simply to relax and take a deep breath, so he will have no reason to try to rush past you. (By the way, the same is true for horses that won’t stand still while you get in the saddle or dismount.)
In regards to good ground manners, leading your horse is not about being mean or rough; it is simply about being consistent and having a clear requirement for where your horse is to walk. Being strict about where your horse walks goes a long way towards making him a good horse in general. By giving your horse clear, consistent direction and a good routine full of creating good habits, your horse will begin to trust you as a leader, and have confidence in what he should doing, and will actually begin to enjoy being a good horse for you.
If you are ever out in Utah, in the Zion National Park or Bryce Canyon National Park area, I hope you will drop in to Rising K Ranch and take a horseback ride with us!