How much of stone-cold basilisk does an opponent have to read in order to trigger the second effect? Is simply reading the first word enough?

Reading any of it counts.

It's finally my birthday! Can I request triva on Stone-Cold Basilisk, Pacifism, or Fog?

Fog is one of only a handful of spells to have its rules text and converted mana cost appear on two other color of cards. For Fog, that’s Darkness (black) and Holy Day (white).Happy Birthday!

Un-ruling: Does Stone-Cold Basilisk have to be on the battlefield for your opponent to become stoned?

Yes. The ability only works on the battlefield.

If an opponent mindslavers you and reads a Stone-Cold Basilisk in your sideboard do they still stoned?

No, the Cold-Stone Basilisk’s ability only works on the battlefield. All effects of permanents only work on the battlefield unless they say otherwise.

You said to eternalcheechako that when someone reads "Stone-Cold Basilisk" they can't respond to the turning to stone ability, why is this? It's worded like a basic triggered ability and even abilities from unhinged must use the stack, as a judge, I don't see why the player can't respond to the ability.

Here’s two answers. One for Melvin and one for Vorthos. Melvin: This effect is like morph in that it doesn’t go on the stack so players do not have the ability to respond to it. Vorthos: You just looked at a basilisk. There’s no time to do anything. As soon as you look, you’re stone.