Just how strict was the wording meant to be for the setup. For instance, in setup it mentions,

"Reveal the leftmost Train Car. Each investigator begins play in that location. (If that location has a 'forced' effect that triggers upon entering that location, ignore it.)"

But there is one train that has a forced trigger for when revealed. Is that one not ignored? Do you still trigger it and spawn a monster on setup since it is not a 'entering' condition?

You are, unfortunately, correct. You do indeed start with a Grappling Horror dragging you down if you begin in the Dining Car.