The first think you should do when you try and break down a visual effects shot is to figure out what elements you need and then plan a way to get them. In the Ice Bucket shot you are going to need at least 3 elements. You need a shot with a clean background, a shot with the guy dumping the water on his head, and a shot with just the bucket falling. This shot also has a moving camera so you have to make some plans so the part where the water falls is shot from very close to the same angle. This can be accomplished buy simply putting some kind of catcher on a stand to move your camera into so that it gets to the same spot for all 3 takes. A simple L braced on a tripod would give you a target to aim for as you are moving the camera.
Now first you shoot the clean plate. You don't need to shoot the whole move here, just shoot from the camera catcher and make sure you have framed things up properly.
Now it's time to do the shot with the guy dumping the water on his head. Rehearse your move a couple of times and then shoot it for real.
The last part of the shooting is the water bucket falling to the ground. This will require two assistants to stand on either side of the actor and hold a the tub of water. You don't care if they are in the shot because you are just going to mask them out anyway. You start with the camera in the catcher and shoot the dropping of the water.
Now that you have your three shots you line them up in After Effects and position each shot so that the background matches the shot with the guy in it as close as you can. Now you do some masking. You'll need the shot of the guy on the bottom layer. You'll need the clean plate in the middle, and the shot of the water and tub falling to the ground at the end. You trim the shot of the tub falling so it matches the shot of the guy dumping the water on himself. Now you start to do some masking. You want to fade out the guy, and maybe even duplicate the layer and roto out just his body so you can scale or wipe him away. The clean plate is also masked to provide a background for the disappearing guy. Then your shot of the tub falling to the ground is masked to hide the two guys holding the bucket and the completion of the shot is added to the end.
If you carefully look at these two still frames from the video you can see the trick. The first still frame has no effects but most of the camera movement stops here as the camera is hit it's target position for the gag.
The second shot, a few frames later reveals all.
Look at the size of the wet pavement. There's way too much water for a single dump. By this time we have cut to the last shot and masked out whatever was used to hold up the tub. The clean Plate is used to replace the background where the two guys were standing or where whatever apparatus they used to drop the tub was placed. Between these two still frames you can see the transition.
The clan plate helps hide the actor as he is wiped from the frame. The shadows do not precisely match why water in this shot because the bottom of the full shot has not yet been replaced with the last shot with the tub falling, and there's another big hint in this still frame. Note the aspect ratio of the shot and especially the top left corner. See the edge of the frame show up? The little curved edge here tells me that the clean plate didn't exactly line up with the other shots at this point.
So there you have it. A minimum of 3 shots, carefully executed, some clever masking and some layering. The falling water hides a multitude of camera errors, the fact that the transition only takes about 3 frames help sell the trick. To get the shots to line up and keep the cuts invisible you would use tracking. I mentioned the aspect ratio of the shot. This tells me that the shot was cropped so the shots would line up. The hardest part is the camera work. The three shots must be all taken from very close to the same position and at the same angle, and the same exposure so that you can hide the cut.
The second video is more of the same. Carefully matched camera angles and some clever masking (roto) and a kid that is willing to go along with dad's funny ideas. Every 'amazing' video effect that you have ever seen starts with careful planning and an understanding of compositing through layers. Every one... There is no magic one click plug-in that will do it for you.
I really appreciate you explaining in great detail how this was accomplished.