Queens are NOT laying 3,000 a day all the time, but here is what Dzierzon said:
“As the queen is capable of adapting the sex of the eggs to the cells, so she is also able to adapt the number of eggs to the requirements of the stock, and to circumstances in general. When a colony is weak and the weather cool and unfavourable she only lays a few hundred eggs daily; but in populous colonies, and when pasture is plentiful, she deposits thousands. Under favourable circumstances a fertile queen lays as many as 3000 eggs a-day; of which any one may convince himself by simply putting a swarm into a hive with empty combs, or inserting empty combs in the brood-nest of a stock, and counting the eggs in the cells some days after.”–Jan Dzierzon, Rational Bee-Keeping, 1882 English edition, Pg 18
“We occasionally read in books on bees, or works on natural history, that the queen in her lifetime lays about 60,000 eggs. Such a statement is simply ridiculous; 600,000 to 1,000,000 would be somewhat nearer the truth; for most queens, in spacious hives and in a favourable season, lay 60,000 eggs in a month. The queen, as a rule, commences laying eggs in February, and continues until September, though not always at the same rate. An especially fertile queen in the four years, which on an average she lives, may thus lay over 1,000,000 eggs. The Author once had a queen fully five years old, which was still remarkably vigorous, and might have lived for another year or two if she had not been destroyed. It is, therefore, quite credible that the age of the queen occasionally extends to seven years, as we are assured by some bee-keepers who have made this observation; yet when we are told that in exceptional cases queens have continued alive for eleven to twelve years, the assertion probably rests on a delusion, or such a case is as rare as that of a man attaining the age of one hundred years or more. There is certainly a great difference among queens as regards fertility; the best mothers are those that lay a great number of eggs and deposit them in the cells regularly, neither laying two eggs in one cell nor missing a cell. With such a queen in the hive the brood is nicely arranged, and much of it hatches simultaneously, thus making it easy for the queen to repeat the operation of depositing eggs when the cells have been emptied. When such is the case the stock will be thriving, its well-being depending chiefly on the queen, who, as it were, is the soul of the hive.”–Jan Dzierzon, Rational Bee-Keeping, 1882 English edition, Pg 18
as far as how many days he said this:
“When the young worker-bee has left the cell — which, reckoning from the egg, will be the case at the end of nineteen days, under favourable circumstances, but generally at the end of twenty to twenty-one days…”–Jan Dzierzon, Rational Bee-Keeping, 1882 English edition, Pg 20
So 19 was the normal number on natural comb. Huber says the same.
As far as how many cells on a frame:
http://bushfarms.com/beesfaqs.htm#cellsonaframe
Deep frame of 5.4mm foundation 7000
Deep frame of 4.9mm foundation 8400
Medium frame of 5.4mm foundation 4620
Medium frame of 4.9mm foundation 5544