It's simple to boot from your OS X installer disk. Just insert that disk. Restart, while holding the C key. You will then boot from that disk. and you can choose Disk Utility from the menus, probably the Utilities menu.
With your other problems, that may not work.
A good next step is to try Disk Warrior.
Sorry to say, I've replaced a LOT of hard drives in MacBooks.
Look forward to whatever happens - then replace the hard drive with new.
The GOOD thing: the MacBook hard drive is simple and quick to replace.
Serendipity is a lucky guess !