Well, the queen may take a few days or months to settle into her nest, once the queen has settles in she may start to lay eggs or not. You can never predict with ants. and lasius niger queens can also control the population of haw many ants she lays.
There is not exact a number on it Kiam, but she can produce a lot of eggs. It depends on the size of the colony and the time of year (winter or summer) also the humidity and temperature.
Normally it does. but if i was you i don't worry about the eggs she is going to lay, just make sure she is in good health and take a good care of your colony. And than the rest will come in no time. (like it does in nature.)
I saw an awesome video on youtube off someone that has this wicked colony built in his house for these kind of ants but i dont know what they are i can send you link if you want ? its really awesome