It's slippery even when not wet. My both horses fall on my driveway when they spooked and turned too fast. So asphalt does have some give (although it's still too hard to do anything besides walk), but it's still dangerous IMO for you and your horse.
You want to keep your heels more down and post by moving your hips, not from the legs as (it looks like) you are doing. Your horse may be pretty unbalanced as well, which doesn't add much to the comfort.
If you have a big exercise ball ($20 in Walmart) you can imitate the trot there (I was taught it at the dressage clinic and it was pretty amazing). Basically you sit on ball with your legs just in same spot and move your hips forward and back to the ball, so the ball would bounce like horse would do.