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Hi, I hope I can provide some insight and help. First, as a storm restoration contractor the insurance company does not usually supply enough materials to get the job done to meet building code, and they are not very responsive when trying to get a supplement, so I can understand why it is taking so long. It happens to me all the time. It is not code to put ice and water on the ridges, if that is what you mean by peaks, and I don't think it is in the valleys in MO, but you could call your local building inspector and just ask. Your shingles should have been installed on a day or two when the temps reached 40 or above as recommended by the manufacturer. With that being said, some of the biggest contractors in town still install them with temps well below that. To me that is a sign of contractor that does not care about quality. If the nails are exposed on the nail line of the shingle they were not installed correctly. Also, when the temps are below 40 the shingles become very brittle, so I'm sure they ripped due to handling in those temps. Those can be replaced very easily if there aren't to many of them. The area of concern is going to be water intrusion. However, I wouldn't worry to much about it on your first year of a new roof. You should be ok, but you need to make sure your issues are resolved.
Any damage to drip edge, fascia, soffit, etc. is also a very easy fix. If they do not fix these any of these items I would highly recommend filing a complaint with the BBB. In the St. Louis, MO area you can call Signature Roofing (573-424-4591) and they will take care of you. He has been trained by Haag Engineering to identify issues with asphalt shingles.
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