Do I have to pay my health insurance bill?

Generally speaking, stopping the payment on any insurance will cancel it. But, some companies have different policies and may require notification. But, usually if they didn't pay anything out, then they usually don't come after you for payment.

I would call and ask if they need a letter or if it'll cancel on its own.

Yes. Insurance is not free just because you don't use it. The price for having insurance is the same whether or not you use it. If you have the insurance, then you have to pay for it, even if you didn't use it.

Using insurance is not the issue, its having the coverage in force. If you paid and they decided not to honor your coverage, would that be ok since they wouldn't be "using" their funds.

When you no longer want your old policy, you need to take action to cancel it. You don't just ignore it.

Yes, unless you notify them that your coverage has been replaced.

Only if you want to maintain coverage.