The boots won't prevent the brushing, just help prevent him from hurting himself when he does it by protecting the areas he's likely to hit.
Some horses will always brush, but it's worth talking to your farrier about it. Sometimes trims can be tweeked to make them track better and prevent them from hitting themselves if the farrier is aware of the problem.
Is your area or the area you ride him particularly dusty? That could cause the coughing problem. I wouldn't worry too much about one or two coughs during warmup, but it shouldn't persist for the entire ride, out of shape or not.