May I ignore the frame damage (or should I get a budget carbon frame) I was happily using my Canonndale F5 Hardtail to commute to work and other places (like shopping, going to a swimming pool and the like) until recently I had a very unfortunate accident with a car. To make a story short, I managed to slightly bent the fork and the frame, where the derailleur is mounted. I would not notice the bent except that I was not able to adjust the gears, if this did not go through the insurance (which demanded a full inspection of the bike). See the pictures below.

The official examiner of the bike dramatically concluded that the bike is unusable. This increased the amount of money I could from the insurance, but the real question is how bad it is? Should I really dump the frame and look for another frame? Or I could try to bent the frame back? (I'll have to replace the fork for sure anyway).

How should one look for a budget frame (are there any types that one should try to avoid?) Being a naive browser I type 'carbon frame' in ebay and being left with hundreds of results. Can someone comment on MUSING X Lite Offroad Mountainbike Frame weighting 1350-grams?

Ultimately I am looking for an inexpensive frame/bike. Key optimization criteria here is the safety feel that one gets during lengthy working, shopping, and swimming hours (i.e., when the bike is being locked outside, passed by some questionnable crowed..). I commute some 10-20km every day.
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.