If someone wants to claim damages to his home against a contractor, does the halacha say somewhere that he should go to the insurance company of
the contractor to file a claim before he takes the contractor to Beis Din?
Or, if he does file a claim for damages and the insurance company says there is no damages, can he then go to Beis Din with his claims? Or does he first need to complain to the insurance company and show the Beis Din what their reasoning was for not paying?
You are allowed to go to the insurance company of the contractor, because that is usual understood in the agreement. I am not aware of any halacha that one must first go to the insurance company before going to bais din.