Can I run a Defrag and close the laptop lid? Or will my computer not defrag when it is in sleep mode? If not, can I turn of the monitor of my laptop when it's not in sleep mode, like, 0 brightness?
The computer won't defrag in sleep mode.
You can set your computer to not go to sleep mode when closing the lid, but I wouldn't recommend that for your use. In fact, I'm not really fond about letting a laptop work when closed, for heat issues, on the monitor.
You can set in the Power Options, in Control Panel, the fact that your monitor will turn off after a few minutes. It will answer to your "0 brightness" wish.
If you're using XP (not sure if it's changed in Vista) you can set up a scheduled task at a time when it is asleep. To have a scheduled task wake the computer you have to choose "Advanced Settings" and check a "Wake Computer" box. I use my netbook as my alarm by scheduling ForWhomTheBellTolls.mp3 to run in the mornings, very convenient. Be sure to harden your OS before using Scheduled Tasks (install all Service Packs and Security Updates).
Then you can get this utility 'http://www.gammadyne.com/cmdline.htm#sleep' and schedule it to put it back to sleep after it's done (when idle).
The best option for you would be to simply automate defrag. Get Diskeeper09 or any other good defragmentation software and set it to defrag in background mode. It gets the job done even as you use your PC. There is no need to defrag manually or schedule tasks.