Can Windows Defrag while Laptop is Closed?

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.

The computer will not defrag when it is in sleep mode or hibernation.

You should be able to adjust your power settings, though, for it not to sleep when the lid is closed.