1A. I clicked "generate coins," now what??
If you happen to be the one to generate coins, they will show up in your interface (you'll get 50 at a time).
HOWEVER: you're very unlikely the be able to generate any coins using the main Bitcoin program. At this point in time, bitcoin mining is a pretty competitive activity. The "professionals" have computing equipment worth thousands of dollars. Even with that, bitcoins may only generated every few weeks. Without top-of-the-line equipment, you'd be lucky to generate any coins, even if you left your computing running 24/7 for a whole year (or even a whole decade).
Because of this, the Generate Coins option is actually being removed from the next version of the Bitcoin program.
Despite the difficulty, mining for bitcoins is still profitable if you're willing to spend the time and money to get serious about it.
1B. Do I ever stop generating blocks/coins??
Your program will keep trying to generate bitcoins until you uncheck the Generate Coins option.
2. How do I open up Bitcoin.conf? I tried following the steps in the wiki page, but no dice.
By default, there is no bitcoin.conf (you have to create it). If you're on windows, it goes in your "%appdata%\Bitcoin" directory.
3. What exactly are these mining operations I keep hearing/reading about??
Because it's currently so rare to generate bitcoins, people get together and help each other do it. Everyone in a "pool" tries to generate bitcoins. If one person in the pool successfully generates some, then the coins are split among all the users in the pool (in proportion to how much help they gave to the pool). Deepbit.net is an example of such a pool.
4. If I put my computer in sleep mode, will bitcoin still run?
No, generating coins will stop and you will be disconnected from the network. However, if someone sends you bitcoins, you will still receive it once you turn your computer back on and rejoin the network.