When you don't have sufficient battery power and power cord stops working, the laptop might abruptly switch off without a proper shutdown and if this happens too often then you have a faulty battery which might have got damaged because of a faulty battery charging power cord. In your case maximum damage you might get is a corrupt Operating system which which won't start up properly. The reason you are getting a blue screen is because there is a conflict of drivers with the hardware.
My suggestions are as follows : Try to insert windows installation dvd, once it boots select repair option and wait for the installation to finish. In this case all your files will be saved in c drive in windows. Old folder,.
If not take technical assistance from the laptop's manufacturer's if the warranty still covers. If not make a visit to your local laptop repair service, there might be some loose connections inside the laptop or there is some hardware issue. Better seek assistance before you make more damage to the laptop.
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