It appears the following states will allow you to use an out of country placard: California, Illinois, Louisiana, Nevada, and Texas. If she'll be visiting other states, you may need to contact the state's DMV for additional information. If you have any questions, don't hesitate to ask.
Since the car is kept at your home, you would need to insure the car in your name and address. You will also need to list your mother as a driver on the policy since she is the owner, and as an additional interest on the policy since she has a financial claim to the car.
Insurance companies don't offer temporary car insurance since policy lengths are typically six months to a year. If you will only be driving the vehicle for one week out of the year and you do not live with your mother, her policy should cover you to drive under permissive use.
The rules for having a non-household member on your policy depends on the specific insurance companies rules and guidelines. Some companies do not like to insure a resident who does not like in the household.
If your mom will be borrowing a car that she does not own, she needs to be listed as a driver on the owner of the car's insurance policy. A non-owners policy is for someone that may drive occasionally but does not have regular access to a specific vehicle.
Since she lives with you and will be driving your car, you will want to add her as a driver to your current policy. This is typically the most cost-effective as well since you already have a policy established.