No — that reputation comes from poor installation. With proper curb-mount flashing, ice & water shield, and step flashing, skylights are 100% waterproof.
Yes — this is the most cost-effective time. The flashing needs to be rebuilt during re-roofing anyway, and a new skylight ensures 20+ years of leak-free performance.
A standard 21x27" skylight delivers the equivalent of 4 windows. The effect on dark kitchens, bathrooms, and hallways is dramatic.