I recall that the key factor for ice to form is depth of the lake (and of course how long its cold for). The top of the body of water cools down to 4 degrees Celsius and then becomes denser and sinks to the bottom of the lake. This cycle then continues with the surface water until all the water of the lake is 4 degrees at which point the water on the top can then freeze. I am assuming that between 4 and 0 degrees the water molecules start to take on a geometry that then allows it to be less dense than liquid water. So deep bodies of water take a long time to freeze because the 4 degree cycle has to repeat so many times.
Our rule of thumb for skating has always been 1.5" if your daring and that 2" is pretty bomber. Of course if there is a huge increase in temp and sun exposure things can get kinda funny...
Rope, ice axe... very smart to bring on a frozen lake. In our fiasco I described where Tom G fell through into cold water, I believe my belt and a piton hammer with a pick end was involved in the haphazard extrication, with me crawling out lizard style toward Tom. As I say, pretty young and dumb, but alive to tell about it.
As to whether I would let my kid go out on such a lake in such a condition, no way. Fortunately, she was and is much smarter about such things than I was in the day.
Another factor to consider is a lot of lakes are fed by groundwater which is typically about 50 degrees or so. Places where groundwater upwells into a lake along the shoreline take much longer to freeze and thaw earlier than elsewhere. Careful examination of ice quality and the surrounding terrain can help to identify this condition. For example where fracture zone intersects lake shore is likely place for this to occur.