Okay, so I've been tinkering away as hard as my feeble mind permits. My toiling has produced a "workable" (although not likable) solution. I can get the character onto a moving vertical platform and (1) keep him there without falling through, and (2) have him perform the "walking animation" when moving right and left.
There are still a few problems related to the Moving Vertical Platforms:
1. The character bounces between 1 to 2 pixels below the platform's top edge. I'm not sure what is causing the bouncing.
2. I had to increase the size of the platform detector from a 1 pixel width to 5 pixels (otherwise, he would fall through the detector). I'm not sure why I had to do this. What this does mean is that the character is able to jump from below the platform and land on it (even though the zenith of the jump was only midway through the center of the platform).
3. I can't get more than one vertical platform on the level. With the horizontal platform code, I can put multiple horizontal platforms in the same level with no problem. With the code that I have right now, multiple vertical moving platforms get all screwed up.
Also, I have a problem with the ladders:
1. If the character jumps off of or leaves a ladder midway through the climb, the "climbing" animation does not turn off. The character then runs around the screen looking like a stumpy troll. I'm not sure why the normal animation is not turning back on.