May I ignore the frame damage (or should I get a budget carbon frame)
My guess is the alignment was like that before the accident and the reason the gears wont adjust is the derailleur hanger is bent, which is a $10 part and/or a visit to the LBS with a tool to fix it. 

Even if the frame has been bent, although less than ideal, it is not a safety issue and would not stop me using it - as long as the derailleur hanger can be aligned correctly. 

However there may be other more critical, but less obvious damage (cracks etc), that you are not aware of, and this be the reason for the insurance write off. 

As for going to carbon frame, for the same money you are almost certainly better off with a steel or alloy frame. Cheap carbon, like everything, is cheap for a good reason. As Carbon is seen as a material with God like qualities by many novices, it commands a price premium based on that perception at the budget end, its until you are paying mid range prices that carbon really shines.