Composite Video Driver Design - PAL
There are hundreds of websites regarding analog video left over from the good old days, and it would be better for you to search for ones that explain things at your level and then come back with more focused questions.

To answer your two questions today:


For a pure monochrome image (black, various shades of gray, white), color burst is not required.  Depending on the country, it might be required in a broadcast signal if the monochrome segment is part of an overall color program, but that does not apply to a non-broadcast test pattern.  Your receiver/monitor should not have a problem with missing burst.
As you noted, a composit video signal is not continuous video information the same way an audio signal is continuous.  Video is punctuated by the H and V sync pulses.  The visual part of the signal is continuous only for the length of one line.  During that time, a continuous analog voltage of 0.714 V (here in the US with NTSC rules) will be interpreted and displayed as pure white.


In its most basic (and slightly non-compliant) form, a white mono video signal needs three voltages:

0.0 V - blanking level (we'll skip the pedestal for now)

-0.286 V - sync tip level for H and V

+0.714 V - white level

http://www.broadcaststore.com/pdf/model/793698/TT148%20-%204053.pdf

https://www.maximintegrated.com/en/app-notes/index.mvp/id/734