1) What are the Good uses for Gold? Powered Rails are a good use. Building a house out of gold is not.
Powered rails are the big one, and a few food items like golden apples, glistering melons, and golden carrots--the latter two are useful for potions. Other than that, they're mostly decorative. You can make armor and tools out of them, but they have terrible durability and so are mostly just for looks.
2) What are the Good uses for Diamond?
Tools and armor, by far. Also enchanting tables.
3) What are the uses of Lapis Lazuli, Emeralds, and Nether Quartz?
Lapis is a blue dye and is also used for enchanting, and the blocks are a nice decorative element. Emeralds are used to trade with villagers. Nether quartz is a fantastic--though expensive--decorative block and is used to make advanced redstone components, comparators being by far the most important.
4) How does Rail work? What can I get away with without Redstone and Powered Rail?
Put a minecart on rails, and you can use it to transport yourself, mobs, (or items, with the use of minecart chests) along them. Hopper minecarts can collect items along the track for you. Powered rails are what make them really useful, though--without them, you can...send things downhill? Or up...if you're willing to push it the entire way.
5) What is the easiest way to kill an Iron Golem? Using only wood, cobblestone, and iron?
Use your iron to make a bucket, and use the bucket to dump lava at its feet. That, or encase it in blocks so it can't move or attack you while you hit it to death, though that takes longer.