Your question makes some broad assumptions, but we will try to answer it.

An insurance company doesn't have any rights, except for those specified in your insurance policy or those specified by your state's insurance regulatory entity.

The rules, procedures and laws regarding vehicle losses vary from state to state and from insurance company to insurance company.

Typically, if your vehicle is totalled then the insurance company can decide if the insurance reimbursement payment should be for the total loss, instead of paying for repairs. If they make payment for a total loss then they will take the car. You may have the option (if the insurance company allows it) to not accept total loss payment, take the money, and pay for the repairs yourself, but if they total it this is not available. You may be able to negotiate for a payment and the car, but only if the cost is less than totalling the vehicle.

An insurance doesn't have the right to cancel your policy because your car is totalled, but there are many reasons a company may cancel a policy. One of those reasons could be caused by (multiple) claims or accident history. They may cancel the policy if you no longer have a car to insure, because it was totalled and you were indemnified (paid). Most state regulations have requirements to protect consumers from unfair cancellation. Typically, if this occurs a company cannot cancel your policy, but they have the option to non-renew your policy. 

When you file your claim with the insurance company, a licensed claims adjuster will provide the details, specific to your situation.