Should obstacle horses be exposed to bicycles, dogs, and vehicles?

Yes — exposure to the types of moving objects that horses regularly encounter in real-world situations is genuinely important for the horse that will be used in any context outside a controlled arena, and doing that exposure in a controlled, safe training setting is far better preparation than encountering these things for the first time while on a trail or at a show where the variables cannot be managed. Bicycles, dogs, and vehicles are among the most commonly encountered moving objects in trail and recreational riding, and each presents a distinct combination of sensory characteristics that requires specific habituation: bicycles move silently and quickly and approach from directions the horse may not anticipate; dogs move unpredictably, make noise, and may approach the horse's legs; vehicles produce noise, vibration, and air movement in addition to visual movement and can appear suddenly at speed. The fundamental introduction protocol for all of these is the same regardless of the specific object: controlled distance, calm and confident handlers, adequate space that allows the horse to move if needed without being trapped, and gradual decrease in distance as the horse demonstrates relaxation at each level. Handlers of the bicycles, dogs, and vehicles during introduction training must understand the protocol — approaching slowly, stopping if the horse shows significant anxiety, giving the horse time to assess before approaching further — and must be prepared to stop and wait rather than continuing the approach if the horse's body language indicates threshold is being approached. Dogs specifically require that their behavior be controlled during the introduction: an unpredictable dog that runs toward the horse or barks suddenly is introducing a different stimulus from a controlled dog that the handler keeps at a consistent distance and prevents from making sudden movements. Each successful exposure to a controlled bicycle, dog, or vehicle in a training context builds the general confidence with these objects that transfers to real-world encounters.