How would discovering a cache of old stock certificates work?

The movie "Blast from the Past" re-used an old story idea to make the main character a millionaire - his dad bought him a small amount of stock in IBM and some other companies when he was born, and now, 35 years later, the stocks are worth a fortune. I'm wondering about the mechanics of this, especially considering the way the stockholders were missing and presumed dead for 35 years.

Let's say you open a book your late grandfather gave you - you're his legal heir - and found the pages packed with 1924 IBM stock certificates that he never mentioned before. Would the fact that the stocks were ignored by the stockholder (and heirs) for some decades cause any difficulties? Would IBM have investigated the "missing" shares? Would they have a record of who owned them? Is it that IBM wouldn't care that no one was voting those shares? If those shares paid dividends over those decades and the owner of the stocks couldn't be located to collect, what would happen to the dividend income? If the stock had split in the intervening years, would each share you held a certificate for be worth two shares?