Interestingly enough, originally it WAS a Mario game, during initial development! Doki Doki Panic started as an engine test for a new Mario game with vertical scrolling in addition to horizontal scrolling. It was swapped to Fuji TV characters shortly after, mid development… and then ironically switched BACK to Mario for the US release!
Yes and yes. I followed the rules as a kid. I played the games with the full rules multiple times even, in defiance of everyone saying that everyone only does so once.
Looking at YouTube playthroughs for nostalgia, it looks like the game rules today have the trap set up to start? When I played, the rules were that you had to build it up slowly over the course of the game, one piece at a time, and only once it was built could it be activated. Yes, I played that way.