What is the ruling on owning a car through a lease-to-own system, and insuring it after its ownership is transferred from the company itself, or owning a private car and insuring its registration?

If the lease is an actual lease with a promise of ownership, then there is no harm in it. As for insuring the car registration (istimara) or insuring with the aforementioned company, if it is cooperative insurance, then there is no objection to it. But if it is commercial insurance, then it is forbidden.