You'll need to have your monitor connected to run the program that wipes out the drive. But after you've got it started you can disconnect the monitor.
Yes you do. You need the monitor on to ensure you have fired up the erasing program correctly. You also need to read any error messages that might pop up. Without eyeballing the task or job completed message at the end you will never know if it was successful.
Depends on the method you're using. I would say probably not if you're zeroing the drive or running dban. Though you won't get any auditory notifications when it's finished, so..
A wipe utility like DBAN will come up with prompts to select a disk, and saying "Do you really want to do this?" Once the wipe is started you can unplug the monitor and just let it run.