if Ethereum is a distributed computing platform
It isn't; it's a replicated/redundant computing platform.
If I make a simple program to get a random number from 1 to 100, or base my program on a randomizer, how can that be repeated in every single node? Shouldn't they give different outputs? How can that be validated?
Let's say I want to make a betting app, based on the random rolling of dice. How can that work?