You don't mention if the flies are doing any damage to the tree? If no damage, I'd leave all as is. Killing the flies will most likely kill the beneficial bugs in the area. Nature has a way of achieving balance between good and bad bugs with the good bugs taking much longer to establish themselves so if you can just let nature do its work. Shoots are produced by the tree on damaged bark. You may have nicked the tree trunk with your lawnmower? The simplest method is to cut them off as close as you can to the trunk(you don't want to leave them as they will take energy from the main tree). I've heard of tar etc that you can paint onto the cut area to prevent further growth but I find it best to cut the shoots with a sharp clean blade and let nature heal the wound. The shoots should stop coming up after a while.