It is my understanding that you can make a claim and the insurance company will pay you what the adjuster(s) estimate the repairs will be (minus deductible). The insurance company will cut you a check for that amount and it is up to you what the money is spent on - repairs, liquor, exhaust mods, whatever you want. If the cost of repairs exceeds the claim payment, it may be difficult to get the overage paid for. YMMV
You are still paying the deductible. The insurance company just subtracts it from the claim payment.
My favorite body shop is on my insurer's preferred list so I let them handle all the details. I just pay the deductible when I pick up the car.
Edit: I do have to request that they use factory parts or the insurance company will try to use after-market parts.