Both are fine, though I don't think I've ever seen a horse wearing a fly bonnet in dressage competition. From the Dressage Equipment handbook
Fly hoods (ear covers) are permitted for competitions held outdoors in order to protect horses from insects. The fly hoods should be discreet and should not cover the horse’s eyes. After completion of the test, the rider or his representative is responsible for removing the fly hood to present to the designated ring steward for inspection. [I doubt you'd have to do this part at a schooling show; when this is required I assume it's to show you don't have ear plugs in underneath it]
I couldn't find a specific mention of half pads, but have personally shown in a square pad with a sheepskin half pad without issue.