When is the cheapest time to travel to Maui?
There is really no 'cheapest' time. The hottest months are Sept - Oct when the Kona winds are not blowing, so, perhaps, that might be the most undesirable time. However, costs remain the same.
When is the best and cheapest time to move to Maui and where is the best city to live in Maui and is the cost of living price balanced with the pay rate?
Aloha: Everyday is a good day to move to Maui. Cheapest... not here. HI is expensive in many ways, but liveable also. Of course, pay is equitable, but comparable to everywhere else in the US (meaning that the economy fluctuates and supports its unique and individual communities). The best city is questionable. Everyone thinks that where they live is the best. I live in Kihei; too touristy sometines, but it's still Maui.