I fox hunt and haul tacked in my straight load when the fixtures are close (10-15 min). I do not put a sheet on, but I do cover my saddle. I don’t haul bridled, but I know plenty of people who do.
I always wait to get to the fixture before putting in studs–but I know people who do not. That seems like asking for trouble, to me–but I’ve never seen a shredded leg from it yet.