How to fix Mountain Lion deadly crashes
Well - if Disk Utility tells you you need to back up your files and erase the drive, that would be my first step at fixing the problem.

It's not possible to tell if you have a hardware or software problem, but once you have backed everything up (or gotten help doing that), then 

Apple usually provides phone support for free within 90 days of purchase, so you might call AppleCare or visit an Apple Store for free service should one be close.

Most people can follow the two articles below to reinstall the OS:


http://www.apple.com/osx/recovery/
PH11273 - OS X Mountain Lion: Erase and reinstall OS X  


Feel free to ask another question if you have a problem within these steps. You could still have a hardware error underlying the issues, but until you try a software fix, it's hard to know that your filesystem isn't just in a bad state where OS X declines to try to fix it.

Your keyboard will be fine for the reinstall, and you don't need bootable media since Internet Recovery will know that Mac shipped with Mountain Lion and offer to re-download the installer when you need it.