It's perfectly legal to carry guns, loaded or unloaded, openly or concealed, at Mt. Charleston and the surrounding National Forest lands. There is a small portion of Red Rock that extends out that way, but it shouldn't be an issue if you're going to the forest lands.
This includes the campgrounds, the trails, the roads, the parking lots, the wilderness, etc.
Where You Can't Carry
No, the state park law doesn't apply to Forest Service land, but the forest service hasn't banned guns anywhere that we're aware of. Red Rock is special, but again, if you're going up Kyle Canyon and recreating in that area, it's not much of an issue for concern.
There is a school and a library up there and they are gun free zones; the school parking lot and grounds are off-limits.
On private property, like the Mt. Charleston lodge, you may be disarmed or asked to leave. If you fail to do so, this is trespassing. At the Lee Canyon ski slopes, we have no information.