Yes, you would pay the contractor the full amount and your deductible, as long as they are completing all of the work. You are being indemnified for the damage, you should not be coming out ahead and making a profit on the claim.
You will receive the payment for actual cash value first, which is the repair cost minus the depreciation and your deductible. If you have replacement cost coverage, you (or your contractor) will notify your insurance company when the work is done and they will release the remaining amount it cost you to replace.
Now that's out of the way, some contractors are willing to negotiate your deductible. With replacement cost coverage, keep in mind that this is not correct in the eyes of the insurance company as you should be footing your portion of the repair cost as agreed upon in the contract, but some people look at it as "What momma dont know wont hurt her." This is completely up to you, as it can be considered fraud. If you have coverage for only actual cash value, the insurance company will pay you for the value of the items now and be done with it so you are free to negotiate as you please.
Hope this helps.