I just found the three numbers that would make the product of their ages 72. Since there is no mention of the same age or the number to use for the sum, I just used the only other clue their was, strawberry shortcake, a childrens cartoon. so the third child was not 18 if the three ages were 18, 4 and 1. so the oldest must still be a child leaving those numbers. But looking at your reasoning about the sum, you had one more clue I couldn't figure out. You sir are correct.

As for #1,you could start by burning the 1st rope at both ends this would only last one half hour and the start burning the second rope from both ends and the middle, there by separating the rope into 4 making it last a total of 15 minutes.