Thanks, glad you enjoy playing! The story mode atm is basically just a difficulty ramp, but we're very excited to incorporate actual story threads/questlines into the game (currently planning on releasing those in v1.6).
You're absolutely correct on the gilded postage, it's a free ticket for any package, nothing else to it. There's no timers in either endless or story mode, and we don't plan on adding them -- chill Papers Please is definitely the vibe :)