No experience with the insurance but here is how you can get the power company there right away.
Send them a certified letter letting them know that the power lines have sagged enough so that vehicle antennas strike the wires creating a SAFETY HAZARD. They will be right out!
-Skip
Be careful about this strategy- they may come right out, cut off the power, and tell you they'll be back after you fix the lines (if it's on your side), or when they have a chance (i.e., "whenever").
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As for the insurance adjusting issue, you should claim for each and every thing which you reasonably can show was damaged or destroyed as a result of the storm. That certainly would include the food in the freezer / fridge, trees, etc. (unless explicitly excluded from the policy's coverage). Do not be shy about claiming legitimate items, and do not be faint of heart about standing up for the appropriate valuation if they try to low-ball you. Most importantly, do not claim anything that does not belong. Your conscience will guide you well here.
But, do not approach it with an automatically-hostile attitude. I know from personal experience that adjusters can go a long way to helping you when they experience a good and honorable attitude.
Who is your carrier? We have USAA, and have experienced very professional and fair adjusting practices in homeowners' insurance claims with them- no "blank check," but no "nickel and diming," either.
Final advice: be very careful about signing off on releases (i.e., "By endorsing and depositing the enclosed, check, Insured waives and releases all claims..."), unless you are quite certain that you have been fully compensated. Good carriers do not require you to release claims to be paid for proven damages- the payment is a defense to a claim for the same damage, and should additional concealed damages arise later on, you would be very disappointed to have released them already.
Again, USAA sends checks (without releases of any kind) based upon estimated damages, and entertains additional claims supported by reasonable facts. Very responsible, and I believe it ultimately reduces fraudulent claims.
My $0.02, worth every penny you paid for it.