I don't own a new LCP, but I plan to. I have owned one previously, a couple of years ago, but sold it.
The LCP is hammer fired, not striker fired (unless that has changed in these new models.) According to what I have read, there was a "concern" (I'm not sure if there were any actual incidents) about a possibly drop/fire in the very early guns, and Ruger recalled and corrected those pistols. (Serial number 370-xxxx made in 2008.) The combination of a very light firing pin and a very strong spring make it very, very unlikely to happen...it would require dropping the gun from a great distance and it to hit squarely on the muzzle. So, odds of it happening are miniscule. Edited to add: while the LCP (I assume still) is carried in a half-cocked state, there is a hammer catch that prevents it from firing until the trigger is pulled. The hammer doesn't have enough inertial force to ignite the primer if it does fall from the half-cocked state, in any normal circumstance.
As for pulling the trigger...the long, heavy trigger pull IS a safety feature. Since I haven't felt the new triggers, I can't say how different they are, but in the model I owned, I couldn't see how it could possibly have gone off accidentally. Pulling the trigger was a deliberate act! I suppose it could be possible that if you rammed the pistol without a holster into a pocket, and the trigger caught on something, and you continued to push with sufficient force, that you could pull the trigger back until the gun fired. I've never tried it to see how much that would take, but I'd guess it would be highly noticeable force and a deep pocket. This is why I always used a good pocket holster with mine, when I carried it.
There is an ongoing debate about carrying a gun without a round chambered. I once had a Bersa BP9CC that had the lightest trigger on it I've ever felt...lighter than my 1911s...and no external safeties at all. I wouldn't carry that gun with a round in the chamber, and in fact sold the gun because I didn't think it was safe to carry. (I love a 1911 trigger because it is so light and crisp, but a 1911 also has a thumb and grip safety.) The question is, if you are ever in a situation where you need your gun to defend yourself, are you going to have the time to chamber a round? And, are you going to be able to do it under pressure? If that is your carry mode, and some people do choose to do so, then you should practice, practice, and practice some more drawing your gun and chambering a round until it becomes second nature...a muscle memory, rather than a conscious act.