Oh Yes They Do! If one party breaches a contract, this does not give the other party the right to breach or rescind it. However, if one of the terms of the contract is that non-payment allows them to cease cover by cancelling the policy, then exercising that option does not amount to "dishonouring" the contract.
I doubt it, except to the extent that you could be forced to pay for the cover for the short period for which it was actually provided, i.e. from renewal to cancellation, perhaps with a reasonable amount added on for admin.