If a player asks if they can use a skill to solve a problem I almost always roll with it, but my rule is that dice rolls are only binding if the DM asked for them. but there are some situations where the roll wouldn't impact the game (e.g. trying to jump to the moon). I think being suspicious of NPCs is reasonable, and I think it's good for the story if some characters are good at sniffing out lies and some trusting souls are just as pure as the driven snow.

As for the players not paying attention, I have been on both sides of the screen and sometimes, though I am trying really hard to pick up what the DM is putting down, the plot is just impenetrable. Still fun, but I have no idea what's going on despite the world-shaking consequences of our actions. Effectively telling a story is hard at the best of times and making it collaborative doesn't do anything to make it easier. If I spin a sprawling epic and the players miss 90% of it, I think that's just the result of a bunch of folks just trying to relax and play a fun game. And if you want your players to remember something, you better be prepared to repeat it a few times.