Must I have my vehicle repaired at the facility referred to me by my insurance company?

R.I.G.L. §27-29-4(15) No! Rhode Island law prohibits insurance companies from requiring that repairs be made to the insured's vehicle at a specified repair center or interfering with the insured's free choice of repair center.

NOT MY FAULT! Do i have to use my insurance company if the accident was not my fault?

R.I. DBR 73 §6D No!"Where liability and damages are reasonably clear, the Insurer shall not recommend that a Third Party Claimant make a claim under his or her own Policy."

If the accident is not your fault the other party's insurer cannot request that you file a claim with your own company.

If I have rental car coverage on my policy will the insurance company pay for the car of my choice?

§ R.I.G.L 27-9.1-4(18) Yes! "No insurance company may interfere with your right to choose a rental company, nor shall it refuse to honor a "direction to pay" executed by the insured; provided, that the rental car company is licensed puruant to R.I.G.L §31-5-33."

This does not, however, mean that the insurance company must provide an identical make/modle vehicle. For example, if your vehicle is a Mercedes SUV than you are entitled to an SUV; however, it could be of any make and modle

Can I request that my vehicle be repaired using only original equipment manufacturer (OEM) parts?

§ 27-10.2-2 (b) Yes and No.
Yes. If your vehicle is less than than thirty (30) months old from the date of manufacture you can request that only OEM parts be used.
No. If your vehicle is older than thirty (30) months the insurance company may stipulate the use of aftermarket (non-OEM) parts

What if my car was totaled; does the insurance company reimburse me for the replacment cost?

R.I DBR 73 §7A(2i) Yes, assuming your policy covers such loss.
"The Insurer may elect to pay a cash settlement equal to the actual cash value of the motor vehicle less any deductible provided in the Policy. In determining the actual cash value of a motor vehicle to settle motor vehicle property damage liability and collision damage claims, Insurers shall use as a guide, the average retail values indicated by the National Automobile Dealers Association official Used Car Guide ("Guide") or some service substantially similar (with appropriate adjustment for such factors as vehicle condition, high and low mileage, accessory options)."