It's better to shoot with both eyes open if you can. Some people simply can't. One of the best shooters I know (my former shooting partner) always closed his non-dominant eye when using his sights.
The reality is that under stress, most people (a) won't use their sights and (b) will have both eyes wide open.
As for the LCP's accuracy, I don't think there is enough data yet to form a cohesive opinion. I'm planning to do some 25yd accuracy testing from a bench with three or four different commercial loads either later this week or next week. I'll be using a Crimson Trace laser to aim, which should help eliminate much of the human factor in terms of the marginal sights that are standard on the LCP. Still, even using just the iron sights, I've been able to keep all my shots in a 6" circle at 25yd offhand. Not spectacular but certainly adequate to the purpose.