If the accident was not your fault, you are entitled to rent a replacement vehicle and the insurance company will reimburse you for it at the end of the rental. They are not responsible to pay for the insurance on it. That coverage extends from your vehicle insurance. In most cases the insurance company will accept the bodyshop invoicing for rentals when you pick up your vehicle.
If you vehicle is a total loss, you are still eligible for a rental vehicle until you are made an offer. Once the offer has been presented you must take the rental vehicle back or you will be responsible for its cost from the date the offer was made.
If you don't carry collision coverage on your vehicle and you are taking a rental vehicle, you must take the full coverage provided by the rental company. you will be expected to pay the daily rate for the additional coverage they provide.
If the accident was your fault and you do not carry loss of use, your insurance company will decline to pay for a rental vehicle.
Some bodyshops may provide you with a courtesy vehicle while its in for repairs but they are not required to do so.
If you do have loss of use then the insurance company will OK the rental to the extent of your loss of use coverage.
ie. Maximum of $50 per day to a maximum of $500 in any one occurrence is a typical coverage.