| I don't know if there are any special laws in Tennessee, so I will answer this based on general knowledge (albeit layman).
The adjuster had me fax over an estimate from one of their approved body shops, which was about $1700.
The adjuster from your company? Or from the company of the person who hit you? Did you take the car to the shop and get the estimate, or did they take it to the shop themselves?
I know that in several states, an insurance company cannot mandate that you go to a specific auto repair facility. Most insurance companies offer "incentives" (IE faster repairs, guaranteed quality, etc) if you go through one of their approved shops, but you are still able to go to any shop you want. Check into your state laws to make sure that this applies to Tennessee, as well.
Then they sent out an appraiser to look at my car.
Generally, the appriaser's job is to look at the car and to use their professional judgement to assess what needs to be repaired. If the insurance company is not happy with the estimate from the autobody shop (IE they think the shop is tacking on stuff that isn't necessary), they'll send an appraiser out for a second opinion.
Does that mean they are going to mail me a check with what the appraiser decides the damages to be, or do they mail me the check in the amount of what the body shop I went to estimated?
Again, in most states, you can choose to go to any shop you want. If the insurance company doesn't like what the shop is doing, they'll let you know and give you the option of paying the difference (IE, if the insurance company thinks the car doesn't need to be completely repainted and that just the front panels need painting, you will be given the option to pay the difference between painting the front panels and painting the entirety of the vehicle). Just remember that the insurance company is responsible for restoring the car to pre-accident condition, or compensating you for the value of the vehicle (should they decide to total it out).
If you want the car repaired and haven't planned on pocketing any of the money, have the insurance deal directly with the autoshop and leave you out of it. That way you don't have to deal with any "supplements" that happen (supplements are the extra charges/work that a body shop has to do to repair a vehicle--things that came up as the repair was made that they hadn't planned on).
Also, are they responsible for a rental car fee for the amount of time the car is estimated to be in the shop, or can they tell me to direct bill it to them (instead of including it in the check to me)?This sort of thing should be covered in your insurance policy. Call your insurance agent (the actual person who sold you the policy, not any of the people you've been dealing with for the accident) to see if car rental is covered. If so, it's usually for up to a certain amount a day (IE $30) with a maximum for a period (IE $900 for 30 days). The insurance adjuster will usually make a reservation for you and you just pick up the car. If you choose a car that's over the daily amount the insurance will pay, you pay the difference (if you want an Escalade, you'll pay the rental company the extra $50 a day) or if you keep the car past the maximum amount, you pay the extra (IE you keep it for 33 days, you pay $90).
If you don't have rental on your insurance, contact the insurance company of the person that caused the accident, and have them work with you.
I hope this helped. | 
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