1. Game Modes.
I would use multiple layouts and event sheets. You can make a base event sheet, and then additional ones for each game mode. Using different layouts will keep from getting cluttered and confusing yourself as well.
2. Dynamic Environments.
Well, for climbing on a wall, I would make the walls you want climbable sprites, and then have events to detect if the player is over these, then activating a group of events which simulate the climbing. Deactivate the group when the player leaves the sprite or dies, etc.
3. Sprites vs TileBG/Box/3Dbox/etc.
Sprites are the main interactive object, yes. But you can still apply movements and all that to Box, etc. TileBG is a cosmetic object mostly.
4. Can someone explain what's going on in this cab?
Will look at it. Someone may beat me to it though.